\documentclass[11pt]{amsart} \usepackage{amsmath,amssymb,amsthm} %\usepackage{version,verbatim} \textwidth=15,5cm \setlength{ \hoffset}{-1cm} \newcommand{\sortnoop}[1]{} \newcommand{\initer}{\setcounter{cste}{\therappel} } \newcommand{\initre}{\setcounter{rappel}{\thecste} } \newcounter{rappel} \newcounter{cste} \newcommand{\cste}{\addtocounter{cste}{1} c_{\thecste}} \newcommand{\mcste}{ c_{\thecste}} \newcommand{\initcste}{\setcounter{cste}{0} } \newcounter{cstei} \newcommand{\cstei}{ c_{\thecstei}} \newcounter{csteii} \newcommand{\csteii}{ c_{\thecsteii}} \newcounter{csteiii} \newcommand{\csteiii}{ c_{\thecsteiii}} \newcounter{csteij} \newcommand{\csteij}{ c_{\thecsteij}} \newcounter{cstej} \newcommand{\cstej}{ c_{\thecstej}} \newcounter{csteji} \newcommand{\csteji}{ c_{\thecsteji}} \newcounter{cstejii} \newcommand{\cstejii}{ c_{\thecstejii}} \newcounter{cstejiii} \newcommand{\cstejiii}{ c_{\thecstejiii}} \newcounter{cstejij} \newcommand{\cstejij}{ c_{\thecstejij}} \newcounter{cstejj} \newcommand{\cstejj}{ c_{\thecstejj}} \newcounter{cstejji} \newcommand{\cstejji}{ c_{\thecstejji}} \newcounter{cstejjii} \newcommand{\cstejjii}{ c_{\thecstejjii}} \newcounter{cstejjiii} \newcommand{\cstejjiii}{ c_{\thecstejjiii}} \newcounter{cstlem} \newcommand{\cstlem}{ c_{\thecstlem}} \newtheorem{theo}{Theorem}[section] \newtheorem{cor}[theo]{Corollary} \newtheorem{prop}[theo]{Proposition} \newtheorem{rem}[theo]{Remark} \newtheorem{lem}[theo]{Lemma} \newtheorem{defi}[theo]{Definition} \newcommand{\cstc}{{\rm c}_0} \newcommand{\delu}{{\rm a}} \newcommand{\csta}{{\delu}_0} %\newcommand{\cstap}{{\delu}'_0} \newcommand{\cstap}{{C}_0} \newcommand{\cstaa}{{\rm b}_1} \newcommand{\cstaaa}{{\rm b'}_1} \newcommand{\cstb}{{\rm b}_0} \newcommand{\cstbb}{{\rm b'}_0} \newcommand{\cstT}{{ \alpha}_0} \newcommand{\Eb}{{\bar{\Bbb E}}} \theoremstyle{plain} \newtheorem{theorem}{Theorem}%[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} %\theoremstyle{remark} \theoremstyle{definition} \newtheorem*{definition}{Definition} \newtheorem*{principle}{Comparison principle} \newtheorem*{remark}{Remark} \newtheorem*{remarks}{Remarks} \newtheorem*{theorem*}{Theorem} \newcommand{\reff}[1]{(\ref{#1})} \newcommand{\findemo}{\hfill $\square$\par \vspace{0.3cm} \noindent} \newcommand{\cst}{c} %\newcommand{\cstc}{{\rm c}_0} %\newcommand{\delu}{{\rm a}} %\newcommand{\csta}{{{\delu}'_0}} %\newcommand{\cstap}{{\delu}_0} %\newcommand{\cstaa}{{\rm b}_1} %\newcommand{\cstb}{{\rm b}_0} %\newcommand{\cstT}{{ \alpha}_0} \newcommand{\rf}{{\rm f}} \newcommand{\rw}{{\rm w}} \newcommand{\W}{{\rm W}} \newcommand{\w}{{\rm w}} \newcommand{\rP}{{\rm P}} \newcommand{\rE}{{\rm E}} \newcommand{\PP}{{\rm \bf P}} \newcommand{\EE}{{\rm \bf E}} %\newcommand{\initer}{\setcounter{cste}{\therappel} } %\newcommand{\initre}{\setcounter{rappel}{\thecste} } %\newcounter{rappel} %\newcounter{cste} %\newcommand{\cste}{\addtocounter{cste}{1} c_{\thecste}} %\newcommand{\mcste}{ c_{\thecste}} %\newcommand{\initcste}{\setcounter{cste}{0} } \newcommand{\ca}{{\mathcal A}} \newcommand{\cb}{{\mathcal B}} \newcommand{\cc}{{\mathcal C}l} \newcommand{\ccc}{{\mathcal C}} \newcommand{\cd}{{\mathcal D}} \newcommand{\ce}{{\mathcal E}} \newcommand{\cf}{{\mathcal F}} \newcommand{\cg}{{\mathcal G}} \newcommand{\ch}{{\mathcal H}} \newcommand{\ci}{{\mathcal I}} \newcommand{\ck}{{\mathcal K}} \newcommand{\cl}{{\mathcal L}} \newcommand{\cn}{{\mathcal N}} \newcommand{\cp}{{\mathcal P}} \newcommand{\cs}{{\mathcal S}} \newcommand{\cw}{{\mathcal W}} \newcommand{\C}{{\mathbb C}} \newcommand{\D}{{\mathbb D}} \newcommand{\E}{{\mathbb E}} \newcommand{\F}{{\mathbb F}} \newcommand{\N}{{\mathbb N}} \renewcommand{\P}{{\mathbb P}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\R}{{\mathbb R}} \newcommand{\Z}{{\mathbb Z}} %pour la biblio% \newcommand{\ddelta}{\Delta} \newcommand{\rbiblio}{\R} \newcommand{\ind}{{\bf 1}} \newcommand{\range}{{\mathcal R}} \newcommand{\demi}{\frac{1}{2}} \newcommand{\sg}{{\rm SG}} \newcommand{\supp}{{\rm supp}\;} \newcommand{\Card}{{\rm Card}\;} \newcommand{\diam}{{\rm diam}\;} \newcommand{\norm}[1]{\mathop{\parallel\! #1 \! \parallel}\nolimits} \newcommand{\val}[1]{\mathop{\left| #1 \right|}\nolimits} \newcommand{\inv}[1]{\mathop{\frac{1}{ #1}}\nolimits} \newcommand{\expp}[1]{\mathop {\mathrm{e}^{ #1}}} \newcommand{\capa}{\mathop{\mathrm{cap}}\nolimits} \newcommand{\capaf}[1]{\mathop{\mathrm{cap}_f( #1 )}\nolimits} \newcommand{\p}[2]{\mathop{\frac{1}{(2\pi #1)^{d/2}}\exp{-\frac{\val{ #2 }^2}{2 #1}}}} \begin{document} \title[Range of super-Brownian motion]{Some properties of the range of super-Brownian motion} \author{Jean-Fran\c{c}ois Delmas} \address{MSRI, 1000 Centennial Drive, Berkeley, CA 94720, U.S.A. \\ and ENPC-CERMICS, 6 av. Blaise Pascal, Champs-sur-Marne, 77455 Marne La Vall\'ee, France.} \email{delmas@msri.org} \thanks{The research was done at the \'Ecole Nationale des Ponts et Chauss\'ees and at MSRI, supported by NSF grant DMS-9701755.} \begin{abstract} We consider a super-Brownian motion $X$. Its canonical measures can be studied through the path-valued process called the Brownian snake. We obtain the limiting behavior of the volume of the $\varepsilon$-neighborhood for the range of the Brownian snake, and as a consequence we derive the analogous result for the range of super-Brownian motion and for the support of the integrated super-Brownian excursion. Then we prove the support of $X_t$ is capacity-equivalent to $[0,1]^2$ in $\R^d$, $d\geq 3$, and the range of $X$, as well as the support of the integrated super-Brownian excursion are capacity-equivalent to $[0,1]^4$ in $\R^d$, $d\geq 5$. \end{abstract} \keywords{Superprocesses, integrated super-Brownian excursion, measure valued process, Brownian snake, hitting probabilities, capacity-equivalence.} \subjclass{60G57, 60J80.} \maketitle \section*{Introduction} Super-Brownian motion, denoted here by $X=(X_t, t\geq 0)$, is a measure-valued process in $\R^d$. It can be obtained as a limit of branching Brownian particle systems. We refer to Dynkin \cite{d:ibmp} for such an approximation in a more general setting. Another way to study super-Brownian motion, is to use the path-valued process, called the Brownian snake, which was introduced by Le Gall \cite{lg:pvmp,lg:pvmppde}. Furthermore this approach allows us to study also the integrated super-Brownian excursion (ISE). This process appears naturally when one consider the limit of rescaled lattice trees in high dimension (see Derbez and Slade \cite{ds:ltsbm,ds:smb}). For every bounded Borel set $A\subset \R^d$, we denote by $A^\varepsilon=\left\{x\in \R^d; d(x,A)\leq \varepsilon \right\}$ and by $\val{A}$ the Lebesgue measure of the set $A$. Recently Tribe \cite{t:rsbm} (see also Perkins \cite{p:smpssbm}) proved a convergence result for the volume of the $\varepsilon$-neighborhood of the support at time $t>0$, $\supp X_t$, of super-Brownian motion in dimension $d\geq 3$. More precisely, Tribe showed that the quantity $\varepsilon^{2-d} \val{\left(\supp X_t\right)^\varepsilon\cap A}$ converges a.s. to a deterministic constant times $\int \ind_A(x) X_t(dx)$. Using results of Le Gall \cite{lg:hppt} on hitting probabilities for the Brownian snake, we give a similar result for the range of the Brownian snake. We then derive an analogous result (theorem \ref{th:rangeX}) for the range of super-Brownian motion after time $t> 0$, $\range_t(X)$ defined as the closure of $\cup_{s\geq t} \supp X_s$. More precisely, we show that there exists a positive constant $\cstap$ depending only on $d$ such that for every Borel set $A\subset \R^d$, $d\geq 4$, for every $t> 0$, we have a.s. \[ \lim_{\varepsilon\rightarrow 0} \varphi_d(\varepsilon)\val{\range_t(X)^\varepsilon\cap A}=\cstap \int_t^\infty ds\;\int \ind_A(z) X_s(dz), \] where $\varphi_4(\varepsilon)=\log (1/\varepsilon)$ and $\varphi_d(\varepsilon)=\varepsilon^{4-d}$ if $d\geq 5$. We also give a similar result for the support of ISE (corollary \ref{cor:cvpsrangeY}). Pemantle and Peres \cite{pp:gwt} defined the notion of capacity-equivalence for two random Borel sets, and later Pemantle and al. \cite{pps:tsbm} showed that the range of Brownian motion in $\R^d$, $d\geq 3$, is capacity-equivalent to $[0,1]^2$. As an application of the previous results , we show (proposition \ref{prop:capaequivX}) that a.s. on $\left\{X_t\neq 0\right\}$, the set $\supp X_t\subset \R^d$, $d\geq 3$, is capacity-equivalent to $[0,1]^2$, and that a.s. the range $\range_t(X)\subset \R^d$ and the support of ISE for $d\geq 5$ are capacity-equivalent to $[0,1]^4$. Let us now describe more precisely the contents of the following sections. In section \ref{sec:setting}, we recall the definition of the path-valued process $W=(W_s,s\geq 0)$ called the Brownian snake. We denote by $\zeta_s$ the lifetime of the path $W_s$. We recall the links between the Brownian snake, super-Brownian motion and ISE. In section \ref{sec:hitting}, we introduce the main tools concerning the Brownian snake. In particular, we consider $T_{(x,\varepsilon)}$ the hitting time for the Brownian snake of $\bar B(x,\varepsilon)$, the closed ball with center $x$ and radius $\varepsilon$: \[ T_{(x,\varepsilon)}=\inf \left\{ s\geq 0; \exists t\in [0,\zeta_s], W_s(t)\in \bar B(x,\varepsilon) \right\}. \] The function $u_\varepsilon(x)=\N_0\left[T_{(x,\varepsilon)}<\infty \right]$, where $\N_0$ is the excursion measure of the Brownian snake away from the trivial path $\rm{0}$, is the maximal nonnegative solution of $\Delta u =4 u^2$ on $\R^d \backslash B(0,\varepsilon)$ (see also Dynkin \cite{d:pacnde}). The study of $\val{\range(W)^\varepsilon\cap A}= \int_A dx\; \ind_{\left\{T_{(x,\varepsilon)}<\infty \right\}}$, where $\range(W)$ is the range of the Brownian snake, relies on the explicit law of the first hitting path $\left(W_{T_{(x,\varepsilon)}}, \zeta_{T_{(x,\varepsilon)}}\right)$ under the excursion measure. This law has been computed by Le Gall \cite{lg:hppt,lg:pccm}. It is closely related to the law of the process $(x_t^\varepsilon, 0\leq t\leq \tau^\varepsilon)$, defined as the unique strong solution of \[ dx^\varepsilon_t=d\beta_t+\frac{\nabla u_\varepsilon(x^\varepsilon_t-x)}{u_\varepsilon(x^\varepsilon_t-x)} dt,\quad \text{for}\quad 0\leq t\leq \tau^\varepsilon, \] where $\beta$ is a Brownian motion in $\R^d$ started at $\beta_0=0$ and $\tau^\varepsilon=\inf\left\{t\geq 0; \val{x^\varepsilon_t-x}=\varepsilon\right\}$. In section \ref{sec:rangeX}, we state the main result on the convergence of the volume of the $\varepsilon$-neighbor\-hood of $\range_t(X)$. The method of the proof is completely different from the one used by Tribe in \cite{t:rsbm}. It is derived from the convergence of the volume of the $\varepsilon$-neighborhood of the range of the Brownian snake in $L^2(\N_{0})$ (proposition \ref{th:erange}). Section \ref{sec:prprop} is devoted to the proof of the latter convergence. The proof of the $L^2(\N_{0})$ convergence is somewhat technical because we need a precise rate of convergence. The derivation of this estimate relies heavily on the explicit law of $\left(W_{T_{(x,\varepsilon)}}, \zeta_{T_{(x,\varepsilon)}}\right)$ under $\N_{0}$. It also depends on precise information on the behavior of the function $u_1$ at infinity. In particular we give the asymptotic expansion of $u_1$ at infinity in the appendix. In section \ref{sec:cap} we prove the results on capacity-equivalence for the support and the range of super-Brownian motion and for the support of ISE. Let $f:[0,\infty )\rightarrow [0,\infty ]$ be a decreasing function. We define the energy of a Radon measure $\nu$ on $\R^d$ with respect to the kernel $f$ by: $ \ci_f(\nu)=\iint f(\val{x-y}) \nu(dx)\nu(dy)$, and the capacity of a set $\Lambda\subset \R^d$ by $ \capaf{\Lambda} =\left[\inf_{\nu(\Lambda)=1} \ci_f(\nu)\right] ^{-1} $. Following the terminology introduced in \cite{pp:gwt}, we say that two sets $\Lambda_1$ and $\Lambda_2$ are capacity-equivalent if there exist two positive constants $c$ and $C$ such that for every kernel $f$, we have \[ c\capaf{\Lambda_1} \leq \capaf{\Lambda_2}\leq C\capaf{\Lambda_1} . \] Proposition \ref{prop:capaequivX} states that a.s. the set $\supp X_t\subset \R^d$, $d\geq 3$, is capacity-equivalent to $[0,1]^2$, and that a.s. the range $\range_t(X)\subset \R^d$, as well as the support of ISE for $d\geq 5$ are capacity-equivalent to $[0,1]^4$. The proof follows the method of \cite{pps:tsbm}. \section{Preliminaries on the Brownian snake and super-Brownian motion} \label{sec:setting} We first introduce some notation. We denote by $(M_f,{\mathcal M}_f)$ the space of all finite measures on $\R^d$, endowed with the topology of weak convergence. We denote by $\cb_{b+} (\R^p)$, respectively $\cb_{b+} (\R^+\times \R^p)$, the set of all real bounded nonnegative measurable functions defined on $\R^p$, respectively on $\R^+\times \R^p$. We also denote by $\cb (\R^p)$ the Borel $\sigma$-field on $\R^p$. For $A\in \cb (\R^p)$, let $\cc (A)=\bar A$ be the closure of $A$. For every measure $\nu\in M_f$, and $f\in \cb_{b+}(\R^d)$, we shall write $\int f(y)\nu(dy) =(\nu,f)$. We also denote by $\supp \nu$ the closed support of the measure $\nu$. If $S$ is a Polish space, we denote by $C(I, S)$ the set of all continuous functions from $I\subset \R$ into $S$. \subsection{The Brownian snake} \label{sec:snake} We recall some facts about the Brownian snake, a path-valued Markov process introduced by Le Gall \cite{lg:pvmp,lg:pvmppde}. A stopped path is a continuous function $\rw :[0,\zeta]\rightarrow \R^d$, where $\zeta=\zeta_{(\rw)}$ is called the lifetime of the path. We shall denote by $\hat \rw $ the end point $\rw(\zeta)$. Let $\cw$ be the space of all stopped paths in $\R^d$. When equipped with the metric \[ d(\rw,\rw')=\val{\zeta_{(\rw)} -\zeta_{(\rw')}}+\sup_{s\geq 0}\val{\rw(s\wedge\zeta_{(\rw)})-\rw'(s\wedge\zeta_{(\rw')})}, \] the space $\cw$ is a Polish space. Let $\rw \in \cw$ and $a,b\geq 0$, such that $a\leq b\wedge\zeta_{(\rw)}$. There exists a unique probability measure on $\cw$ denoted by $Q_{a,b}^\rw(d\rw')$ such that: \begin{itemize} \item[(i)] $\zeta_{(\rw')}=b$, $Q_{a,b}^\rw(d\rw')$-a.s. \item[(ii)] $\rw'(t)=\rw(t)$ for every $0\leq t\leq a$, $Q_{a,b}^\rw(d\rw')$-a.s. \item[(iii)] The law of $\left(\rw'(t+a), 0\leq t\leq b-a \right)$ under $Q_{a,b}^\rw(d\rw')$ is the law of Brownian motion in $\R^d$ started at $\rw (a)$ and stopped at time $b-a$. \end{itemize} We shall also consider $Q_{a,b}^\rw(d\rw')$ as a probability on the space %of continuous functions $C([0,b],\R^d)$. We set $\cw_x=\left\{\rw \in \cw; \rw(0)=x \right\}$ for $x\in \R^d$. Let $\rw \in \cw_x$. We restate theorem 1.1 from \cite{lg:pvmp}: \begin{theo}[Le Gall] \label{th:serpentlg} There exists a continuous strong Markov process with values in $\cw_x$, $W=(W_s,s\geq 0)$, whose law is characterized by the following two properties. \begin{itemize} \item[(i)] The lifetime process $\zeta=\left(\zeta_s=\zeta_{(W_s)},s\geq 0\right)$ is a reflecting Brownian motion in $\R^+$.% which starts at $\zeta_0=\zeta_{(\rw)}$. \item[(ii)] Conditionally given $\left(\zeta_s,s\geq 0\right)$, the process $\left(W_s,s\geq 0\right)$ is a time-inhomogeneous continuous Markov process, % which starts at %$W_0=\rw$ and whose transition kernel between times $s$ and $s'\geq s$ is \[ P_{s,s'}(\rw,d\rw')=Q_{m(s,s'),\zeta_{s'}}^\rw(d\rw'), \] where $m(s,s'):=\inf_{r\in [s,s']}\zeta_r$. \end{itemize} \end{theo} From now on we shall consider the canonical realization of the process $W$ defined on the space $C(\R^+,\cw_x)$. The law of $W$ started at $\rw$ is denoted by $\ce_{\rw}$. We will use the following consequence of (ii): outside a $\ce_\rw$-negligible set, for every $s'>s$, one has $W_s(t)=W_{s'}(t)$ for every $t\in [0,m(s,s')]$. We shall write $\ce^*_\rw$ for the law of the process $W$ killed when its lifetime reaches zero. The distribution of $W$ under $\ce^*_\rw$ can be characterized as in theorem \ref{th:serpentlg}, except that its lifetime process is distributed as a linear Brownian motion killed at its first hitting time of $\{0\}$. The state space for $\left(W,\ce^*_\rw\right)$ is the space $\cw_x^*=\cw_x\cup \partial$, where $\partial$ is a cemetery point. The trivial path $\textbf{x}$ such that $\zeta_{(\textbf{x})}=0$, $\textbf{x} (0)=x$ is clearly a regular point for the process $\left(W, \ce_{\rw}\right)$. Following \cite{b:emp} chapter 3, we can consider the excursion measure, $\N_x$, outside $\left\{\textbf{x}\right\}$. The distribution of $W$ under $\N_x$ can be characterized as in theorem \ref{th:serpentlg}, except that now the lifetime process $\zeta$ is distributed according to It\^o measure of positive excursions of linear Brownian motion. We normalize $\N_x$ so that, for every $\varepsilon>0$, \[ \N_x\left[\sup_{s\geq 0}\zeta_s>\varepsilon\right]=\inv{2\varepsilon}. \] The Brownian snake enjoys a scaling property: if $\lambda>0$, the law of the process $W^{(\lambda)}_s(t)= \lambda^{-1} W_{\lambda^4 s} (\lambda^2 t)$ under $\N_x$ is $\lambda^{-2} \N_{\lambda^{-1}x}$. We recall the strong Markov property for the snake under $\N_x$ (see \cite{lg:pvmppde}). Let $T$ be a stopping time of the natural filtration $\cf^W$ of the process $W$. Assume $T>0$ $\N_x$-a.e., and let $F$, $H$ nonnegative measurable functionals on $C(\R^+,\cw_x^*)$ such that $F$ is $\cf_T^W$ measurable. Then if $\theta$ denotes the usual shift operator, we have \[ \N_x\left[T<\infty ; F\:\cdot\: H\circ \theta_T\right] = \N_x\left[T<\infty ;F\:\cdot\: \ce_{W_T}^*\left[H\right]\right]. \] Let $\sigma=\inf\left\{s>0;\zeta_s=0\right\}$ denote the duration of the excursion of $\zeta$ under $\N_x$. The range $\range =\range (W)$ of $W$ is defined under $\N_x$ by \[ \range=\left\{W_s(t); 0\leq t\leq \zeta_s, 0\leq s \leq \sigma\right\} = \left\{\hat W_s; 0\leq s \leq \sigma \right\}. \] For every nonnegative measurable function $F$ on $\cw_x^*$, we have \begin{equation*} %\label{eq:NGds} \N_x\left[\int_0^\sigma F(W_s,\zeta_s) ds \right] = \int_0^\infty \E_x\left[F(\beta_{[0,t]},t)\right] dt, \end{equation*} where $\beta_{[0,t]}$ is under $\P_x$ the restriction to $[0,t]$ of a Brownian motion in $\R^d$ started at $\beta_0=x$. Now consider under $\N_x$ the continuous version $\left(l^t_s,t> 0,s\geq 0\right)$ of the local time of $\zeta$ at level $t$ and time $s$. We define a measure valued process $Y$ on $\R^d$ by setting for every $t>0$, for every $\varphi\in \cb_{b+}(\R^d)$, \[ (Y_t,\varphi)=\int_0^\sigma dl_s^t \; \varphi(\hat W_s). \] We shall sometimes write $Y_t(W)$ to recall that $Y_t$ is a function of the Brownian snake. From the joint continuity of the local time and the continuity of the map $s\mapsto \hat W_s$, we get that $\N_x$-a.e., the process $Y$ is continuous on $(0,\infty )$ for the Prohorov distance on $M_f$. Let $\varphi\in \cb_{b+}(\R^d)$. We define on $\R^+\times \R^d$ the function $v(t,x)=\N_x\left[1-\exp{-(Y_t,\varphi)}\right]$, if $t>0$, and $v(0,x)= \varphi(x)$. We will write $v(t)$ for the function $v(t,\cdot)$. We recall that the function $v$ is the unique nonnegative measurable solution of the integral functional equation \begin{equation} \label{eq:integrale} v(t)+2\int_0^t ds\; P_s\left[v(t-s)^2\right]=J(t)\qquad t\geq 0, \end{equation} where $J(t,x) =P_t [\varphi](x)$, and $(P_t,t\geq 0)$ is the Brownian semi-group in $\R^d$. A few other remarks on the solution of (\ref{eq:integrale}) are presented in section \ref{sec:annexemo} below. \subsection{Super-Brownian motion and ISE} Let us now recall the definition of super-Brownian motion %(see \cite{d:bpss,d:ibmp}) and its connection with the Brownian snake. %(see \cite{lg:pvmp,lg:pvmppde}) The second part of the next theorem is lemma 4.1 from \cite{d:bpss}. Let $\nu\in M_f$. \begin{theo} %Let $\nu\in M_f$. There exists a continuous strong Markov process $X=\left(X_s,s\geq 0\right)$ defined on the canonical space $C(\R^+, M_f)$, whose law is characterized by the two following properties under $ \P^X_\nu$. \begin{itemize} \item[(i)] $X_0=\nu$, $\P^X_\nu$-a.s. \item[(ii)] For every $\varphi\in \cb_{b+}(\R^d)$, $t\geq s>0$, we have \begin{equation*} %\label{eq:LaplaceX} \E_\nu^X\left[\exp{\left[-(X_t,\varphi)\right]}\mid \sigma(X_u,0\leq u\leq s)\right] = \exp{\left[-(X_s,v(t-s))\right]} , \end{equation*} where the function $v$ is the unique nonnegative solution of (\ref{eq:integrale}) with $J(t)=P_t[\varphi]$. \end{itemize} Furthermore, for every integer $m\geq 1$, $t_m>\cdots>t_1\geq 0$, $\varphi_1,\ldots,\varphi_m \in \cb_{b+}(\R^d)$, we have \begin{equation} \label{eq:fLaplaceX} \E_\nu^X\left[\exp{\left[-\sum_{\left\{i;t_i\leq t\right\}}(X_{t-t_i},\varphi_i)\right]}\right] = \exp{\left[-(\nu,v(t))\right]} , \end{equation} where $v$ is the unique nonnegative solution to the integral equation (\ref{eq:integrale}) with right-hand side $J(t)=\sum_{\left\{i;t_i\leq t\right\}}P_{t-t_i}[\varphi_i]$. \end{theo} \begin{theo}[Le Gall \cite{lg:pvmp,lg:pvmppde}] \label{th:XY} Let $\sum_{i\in I} \delta_{W^i}$ be a Poisson measure on $C(\R^+,\cw)$ with intensity $\int \nu(dx) \N_x[\cdot]$, then the process $Z$ defined by $Z_0=\nu$ and $Z_t=\sum_{i\in I}Y_t(W^i)$ if $ t>0$, is distributed according to $\P_\nu^X$. \end{theo} We deduce from the normalization of $\N_x$ that $\N_x\left[Y_t\neq 0\right]=1/2t<\infty $. This implies that for every $t>0$, there is only a finite number of indices $i\in I$ such that the process $(Y_s(W^i), s\geq t )$ is nonzero. We now recall the connection between ISE and Brownian snake. There exists a unique collection $\left(\N^{(r)}_0, r>0\right)$ of probability measure on $C(\R^+,\cw_0^*)$ such that: \begin{enumerate} \item For every $r>0$, $\N^{(r)}_0[\sigma=r]=1$. \item For every $\lambda>0$, $r>0$, $F$, nonnegative measurable functional on $C(\R^+,\cw_0^*)$, \[ \N^{(r)}_0\left[F(W^{(\lambda)})\right]=\N^{(\lambda^{-4} r)}_0\left[F(W)\right]. \] \item For every nonnegative measurable functional $F$ on $C(\R^+,\cw_0^*)$, \begin{equation} \label{eq:Nnormal} \N_0[F]=\inv{\sqrt{2\pi}} \int_0^\infty dr\; r^{-3/2} \N^{(r)}_0[F]. \end{equation} \end{enumerate} The measurability of the mapping $r\mapsto \N^{(r)}_0[F]$ follows from the scaling property 2. Under $\N^{(1)}_0$, the distribution of $W$ is characterized as in theorem \ref{th:serpentlg}, except that the lifetime process is distributed according to the normalized It\^o measure. The law of the ISE is the law of the continuous tree associated to $\sqrt{2}W$, under $\N^{(1)}_0$ (see corollary 4 in \cite{lg:urtbe} and \cite{a:tbm}). In particular the law of the support of ISE is the law of $\sqrt{2}\range$ under $\N^{(1)}_0$, where we set $\lambda A=\{x; \lambda^{-1} x\in A\}$. \subsection{Hitting probabilities for the Brownian snake} \label{sec:hitting} We now recall a few results from \cite{lg:hppt}. Let $\rw \in \cw\cup C(\R^+,\R^d)$, we introduce the first hitting time of $A\in \cb(\R^d)$: \[ \tau_{A}(\rw)=\inf \left\{t\geq 0; \rw(t)\in A \right\}, \] with the usual convention $\inf \emptyset =\infty $. We omit $\rw$ when there is no risk of confusion. Consider the Brownian snake $W$, and set \[ T_{(y,\varepsilon)}=\inf \left\{s\geq 0; \exists t\in [0,\zeta_s], W_s(t)\in \bar B(y,\varepsilon) \right\}, \] where $B(y,\varepsilon)$ is the open ball in $\R^d$ centered at $y$ with radius $\varepsilon> 0$, and $\bar B(y,\varepsilon)$ its closure. We know from \cite{lg:pvmppde} that the function defined on $\R^d\backslash \bar B(0,\varepsilon)$, \[ u_\varepsilon(y):=\N_0\left[T_{(y,\varepsilon)}<\infty \right]=\N_0\left[\range \cap \bar B{(y,\varepsilon)}\neq \emptyset \right]=\N_{-y}\left[\range \cap \bar B{(0,\varepsilon)}\neq \emptyset \right] , \] is the maximal nonnegative solution on $\R^d\backslash \bar B(0,\varepsilon)$ of \[ \Delta u= 4u^2. \] This result was first proved in a more general setting by Dynkin \cite{d:pacnde} in terms of superprocesses. The function $u_\varepsilon$ is strictly positive on $\R^d\backslash \bar B(0,\varepsilon)$. For every $y_0\in \partial B(0,\varepsilon)$, we have \[ \lim_{y\in \bar B(0,\varepsilon)^c; y\rightarrow y_0} u_\varepsilon(y)=\infty . \] Scaling and symmetry arguments show that for every $y\in \R^d\backslash\bar B(0,\varepsilon)$, \begin{equation} \label{eq:uscale} u_\varepsilon(y)=\varepsilon^{-2} u_1\left(\frac{\val{y}}{\varepsilon}\right), \end{equation} where the function $u_1(r)$, $r\in (1,\infty )$ is the maximal nonnegative solution on $(1,\infty )$ of \[ u''_1(r)+\frac{d-1}{r} u'_1(r) =4 u^2_1(r). \] It is easy to see that the function $u_1$ is decreasing. In section \ref{sec:annexeu} we give the asymptotic expansion of $u_1$ at infinity. We give the following result on the probability of the event $\left\{T_{(y,\varepsilon)}<\infty \right\}$ (see lemma 2.1 of \cite{lg:hppt}). Assume $x_0\not \in \bar B(y,\varepsilon)$. Then $\N_{x_0}$-a.e. for every $T \geq 0$, we have \begin{align} \label{eq:hitting} \ce^*_{W_T} \left[T_{(y,\varepsilon)}<\infty \right] &=2\int_0^{\zeta_T\wedge \tau_{B(y,\varepsilon)}(W_T)} dt\;u_\varepsilon(W_{T}(t)-y) ) \expp{\left[-2\int_0^t u_\varepsilon(W_{T}(s)-y) ds \right]}\\ \nonumber &=1- \exp{\left[-2\int_0^{\zeta_T\wedge \tau_{B(y,\varepsilon)}(W_T)} u_\varepsilon(W_{T}(s)-y) ds \right]}. \end{align} Let $x_0,x\in \R^d$. We will now describe the law of $W_{T_{(x,\varepsilon)}}$ under $\N_{x_0}[\cdot\mid T_{(x,\varepsilon)}<\infty ]$. First of all we denote by $\beta$ a Brownian motion in $\R^d$ started at $x_0$ under $\P_{x_0}$. Assume $x_0\not \in \bar B(x,\varepsilon )$. Corollary 2.3 from \cite{lg:hppt} ensures that there exists $\P_{x_0}$-a.s. a unique continuous process $x^\varepsilon= (x^\varepsilon_t, 0\leq t\leq \tau^\varepsilon)$ taking values in $\R^d$ such that for every $\eta\in(0, \val{x-x_0} -\varepsilon)$, for every $t\leq \tau_\eta^\varepsilon=\inf \left\{s\geq 0; \val{x^\varepsilon_s-x} \leq \varepsilon+ \eta \right\}$, \[ x^\varepsilon_t=\beta_t + \int_0^{t} \;\frac{\nabla u_\varepsilon (x^\varepsilon_s-x)}{u_\varepsilon (x^\varepsilon_s-x)} ds, \] furthermore, $\P_{x_0}$-a.s. $\tau^\varepsilon=\lim_{\eta\rightarrow 0} \tau_\eta^\varepsilon<\infty $ and $\val{x^\varepsilon_{\tau^\varepsilon} -x}= \varepsilon$. We also recall that thanks to Girsanov's theorem, we have for every nonnegative measurable function $F$ on $C([0,t], \R^d)$ \begin{multline*} \E_{x_0} \left[\tau^\varepsilon>t; F\left(x^\varepsilon_{[0,t]}\right)\right] \\ =\E_{x_0}\left[\tau_{B(x,\varepsilon)}(\beta)>t;F\left(\beta_{[0,t]} \right)\frac{u_\varepsilon(\beta_t-x)}{u_\varepsilon(x_0-x)}\exp{\left[-2\int_0^t u_\varepsilon(\beta_s-x) ds \right]} \right], \end{multline*} where $x^\varepsilon_{[0,t]}$ and $\beta_{[0,t]}$ are the restriction of $x^\varepsilon$ and $\beta$ to $[0,t]$. The law of $x^\varepsilon$ under $\P_{x_0}$ can be interpreted as a probability measure on $\cw_{x_0}^*$. Consider the closed set \[ A=\left\{\rw\in \cw_{x_0}^*; \tau_{\bar B(x,\varepsilon)}(\rw)<\infty \right\}. \] It has been proved in \cite{lg:hppt} that its capacitary measure with respect to the Brownian snake with initial point $x_0$ is exactly $u_\varepsilon(x_0-x)$ times the law of $x^\varepsilon$ under $\P_{x_0}$. It is not hard to check however that the normalized capacitary measure can be interpreted as the hitting distribution under $\N_{x_0}$ (cf \cite{lg:pccm}, this is proved in a way similar to the classical interpretation of the capacitary measure as a last exit distribution, see e.g. Port and Stone \cite{ps:bm}). Thus we deduce that for every nonnegative measurable function $F$ on $\cw_{x_0}^*$, we have \begin{equation*} %\label{eq:Nnu} \N_{x_0}\left[T_{(x,\varepsilon)}<\infty ; F(W_{T_{(x,\varepsilon)}},\zeta_{T_{(x,\varepsilon)}})\right] =u_\varepsilon(x_0-x)\E_{x_0} \left[F(x^\varepsilon,\tau^\varepsilon)\right]. \end{equation*} Hence for every $t\geq 0$, and every nonnegative measurable function $F$ on $C([0,t], \R^d)$, we have \begin{multline} \N_{x_0}\left[T_{(x,\varepsilon)}<\infty ; \zeta_{T_{(x,\varepsilon)}}>t; F\left((W_{T_{(x,\varepsilon)}}(s),s\in [0,t])\right)\right]\\ \label{eq:Nbeta} =\E_{x_0}\left[\tau_{B(x,\varepsilon)}>t; F\left(\beta_{[0,t]}\right){u_\varepsilon(\beta_t-x)}\exp{\left[-2\int_0^t u_\varepsilon(\beta_s-x) ds \right]} \right]. \end{multline} Finally we shall use the following inequality, that can be derived from the Feynman-Kac formula (use the fact that $u_\varepsilon$ solves $\Delta u=4u_\varepsilon u$) \begin{equation} \label{eq:fk} \hspace{-0.5cm}u_\varepsilon(x)\geq 2\E_0 \left[ \int_0^{\tau_{B(x,\varepsilon)} }dt\;\; {u_\varepsilon(\beta_t-x)}^2\exp{\left[-4\int_0^t u_\varepsilon(\beta_s-x) ds \right]}\right]. \end{equation} There is in fact equality in \reff{eq:fk} (see the remark on page 293 of \cite{lg:hppt}). \section{A property of the range of super-Brownian motion} \label{sec:rangeX} \initcste \noindent For $A\in \cb(\R^d)$, $\varepsilon>0$, we set $ A^\varepsilon:=\left\{x\in \R^d; d(x,A)\leq \varepsilon \right\}$, with $d(x,A)=\inf \left\{ \val{x-y}; y\in A\right\}$. We will write $\val{A}$ for the Lebesgue measure of $A$. We also set \[ \cstap=\csta 2\pi^{d/2} \Gamma([d-2]/2)^{-1}, \] where the constant $\csta$ is defined in lemma \ref{lem:a0} (see also the remark below the lemma). We set $\range_t(X)=\cc \left(\bigcup _{s\geq t} \supp X_s\right)$. Let $\varphi_d(\varepsilon)=\varepsilon^{4-d}$ if $d\geq 5$ and $\varphi_4(\varepsilon)=\log (1/\varepsilon)$ for $\varepsilon>0$. \begin{theo} \label{th:rangeX} Let $\nu\in M_f$. For every Borel set $A\subset \R^d$, $d\geq 4$, for every $t>0$, $\P_\nu^X$-a.s. \begin{equation} \label{eq:rangeX} \lim_{\varepsilon\rightarrow 0} \varphi_d(\varepsilon)\val{\range_t(X)^\varepsilon\cap A}=\cstap \int_t^\infty ds\;(X_s,\ind_A). \end{equation} If there exists $\rho<4$ such that $\lim_{\varepsilon\rightarrow 0} \varepsilon^{\rho-d}\val{(\supp \nu)^\varepsilon}=0$ then \reff{eq:rangeX} holds with $t=0$. \end{theo} Let $K$ a compact subset of $\R^d$. We consider the measure $\phi(K)$ defined by $\phi(K)(A)=\val{K\cap A}$. Since the set $\range_t(X)$ is compact for $t>0$, the theorem implies that a.s. the sequence of measures $(\varphi_d(\varepsilon) \phi(\range_t(X)^\varepsilon), \varepsilon>0)$ converges weakly to $\cstap \int_t^\infty ds\;(X_s,\ind_A)$. Let us recall the main theorem of \cite{t:rsbm} (see also \cite{p:smpssbm}). \begin{theo}[Tribe] \label{th:tribe} Let $A$ a bounded Borel set in $\R^d$, $d\geq 3$. Fix $t>0$ and $\nu\in M_f$. Then there exists a positive constant $\cstT$ depending only on $d$ such that \[ \lim_{\varepsilon\rightarrow 0} \varepsilon^{2-d}\val{\left(\supp X_t\right)^\varepsilon\cap A} =\cstT (X_t,\ind_A), \] where the convergence holds $\P^X_\nu$-a.s. and in $L^2(\P^X_\nu)$. \end{theo} We shall deduce theorem \ref{th:rangeX} from the next proposition on the range of the Brownian snake, whose proof will be given in the next section. For $\theta\in (0,1)$, we set $h_{d,\theta}(\varepsilon)=\varepsilon^{1-\theta}$ if $d\geq 5$ and $h_{4,\theta}(\varepsilon)=\log (1/\varepsilon)^{-1/\theta}$ for $\varepsilon\in (0,1)$. For short we will write $h_d$ for $h_{d,\theta}$. \begin{prop} \label{th:erange} Let $d\geq 4$. For every $\theta\in (0,1/d)$ and every $R_0>0$, there exists a constant $\kappa=\kappa(\theta)>0$ and $\varepsilon_0>0$ such that for every $\varepsilon\in (0,\varepsilon_0]$, for every $x_0$ with $\val{x_0}\leq R_0$, and every Borel set $A\subset \bar B(0,R_0)$, we have \[ \val{\N_{x_0}\left[\varphi_d(\varepsilon)\val{\range(W)^\varepsilon\cap A\cap \bar B(x_0,h_d(\varepsilon))^c }-\cstap \int_0^\infty ds\;(Y_s,\ind_A)\right]}\leq h_d(\varepsilon)^{\kappa/2}, \] and \[ \N_{x_0}\left[\left[\varphi_d(\varepsilon) \val{\range(W)^\varepsilon\cap A\cap \bar B(x_0,h_d(\varepsilon))^c }-\cstap \int_0^\infty ds\;(Y_s,\ind_A)\right]^2\right]\leq h_d(\varepsilon)^\kappa. \] \end{prop} \noindent \textbf{Remark}. We have trivially $B(x_0,\varepsilon)\subset \range(W)^\varepsilon$, $\N_{x_0}$-a.e. Since $\N_{x_0}$ is an infinite measure, $\N_{x_0} \left[ \val{\range(W)^\varepsilon\cap B(x_0,\delta)}\right]=\infty $ for every $\varepsilon,\delta>0$. This is the reason why we consider $A\cap \bar B(x_0,h_d(\varepsilon))^c $ rather than $A$ in the previous proposition. We first give a consequence of this proposition. \begin{cor} \label{cor:cvpsrangeY} Let $d\geq 4$. For every Borel set $A\subset \R^d$, $\N_{x_0}$-a.e., we have \[ \lim_{\varepsilon\rightarrow 0} \varphi_d(\varepsilon)\val{\range(W)^\varepsilon \cap A}=\cstap \int_0^\infty ds\; (Y_s,\ind_A). \] The results holds $\N^{(1)}_0$-a.s. if $\val {\partial A}=0$. \end{cor} \noindent \textbf{Proof} of corollary \ref{cor:cvpsrangeY}. Since $\N_{x_0}$-a.e. the range $\range(W)$ is bounded, we only need to consider a bounded Borel set $A$. Let $\kappa>0$ be fixed as in proposition \ref{th:erange}. Let $\varepsilon_n$ such that $h_d(\varepsilon_n)=n^{-2/\kappa}$ for $n\geq 1$. Using the Borel-Cantelli lemma and the second upper bound of proposition \ref{th:erange}, we get that the sequence $\left(\varphi_d(\varepsilon_n)\val{\range(W)^{\varepsilon_n} \cap A},n\geq 1\right)$ converges $\N_{x_0}$-a.e. to $\cstap \int_0^\infty ds\; (Y_s,\ind_A)$. But for $\varepsilon'\leq \varepsilon$, since $\range(W)^{\varepsilon'} \subset \range(W)^\varepsilon$, we have \[ \varphi_d({\varepsilon'}) \val{\range(W)^{\varepsilon'}\cap A} \leq \varphi_d({\varepsilon}) \val{\range(W)^\varepsilon \cap A}\varphi_d({\varepsilon'})/\varphi_d({\varepsilon}). \] A monotonicity argument using the fact that $\varphi_d({\varepsilon_{n+1}})/\varphi_d({\varepsilon_n})$ converges to $1$, completes the proof of the first part. The above result implies that $\N_0$-a.e. the sequence of measures $(\varphi_d(\varepsilon) \phi(\range(W)^\varepsilon),\varepsilon>0)$ converges weakly to $\cstap \int_0^\infty ds\; Y_s$. Using \reff{eq:Nnormal} we see this convergence also holds $dr$-a.e. $\N^{(r)}_0$-a.s. By the scaling property the Brownian snake and the family $(\N^{(r)}_0, r>0)$, we get this convergence holds $\N^{(1)}_0$-a.s. Thus we have for every Borel set $A\subset \R^d$, $\N^{(1)}_0$-a.s. \begin{multline*} \cstap \int_0^\infty ds\; (Y_s,\ind_{\text{Int}(A)})\leq \liminf_{\varepsilon\rightarrow 0} \varphi_d(\varepsilon) \val{\range(W)^\varepsilon\cap A}\\ \leq \limsup_{\varepsilon\rightarrow 0} \varphi_d(\varepsilon) \val{\range(W)^\varepsilon\cap A} \leq \cstap \int_0^\infty ds\; (Y_s,\ind_{\bar A}), \end{multline*} where Int$(A)$ denotes the interior of $A$. To prove the second part of the corollary we just need to check that if $\val{\partial A}=0$ then $\int_0^\infty ds\; (Y_s,\ind_{\text{Int}(A)})=\int_0^\infty ds\; (Y_s,\ind_{\bar A})$. It is enough to prove that $\val{A}=0$ implies $\int_0^\infty ds\; (Y_s,\ind_{A})=0$ $\N^{(1)}_0$-a.s. Conditioning by the lifetime process, we get \[ \N^{(1)}_0\left[\int_0^\infty ds\; (Y_s,\ind_{ A})\right]=\N^{(1)}_0\left[\int_0^1 dt\; \ind_{A}(\hat W_t)\right]=\int_0^1 dt \;\N^{(1)}_0\left[P_{\zeta_t}[\ind_A](0)\right]. \] This is equal to zero if $\val{A}=0$. This ends the proof of the second part of the corollary. \findemo As a byproduct of the proof we get that $\N_{x_0}$-a.e. and $\N^{(1)}_0$-a.s. the sequence of measures $(\varphi_d(\varepsilon) \phi(\range(W)^\varepsilon),\varepsilon>0)$ converges weakly to $\cstap \int_0^\infty ds\; Y_s$. We first state some straightforward consequences of \reff {eq:uscale} and lemma \ref{lem:a0}. We say that $\varepsilon_0>0$ satisfies the condition (C) if $\varepsilon_0^{-\theta}\geq 4/3$ if $d\geq 5$ or $\log (1/\varepsilon_0)\geq 4 \log (2/\theta)/\theta$ if $d=4$. For $\theta\in (0,1/4)$ this implies that for $\varepsilon\in (0,\varepsilon_0)$, $h_4(\varepsilon)/\varepsilon\geq 4/3$ and that \begin{equation} \label{eq:lolo} \log (\log (1/\varepsilon))/[\theta \log (1/\varepsilon)]\leq 1/2. \end{equation} For $d\geq 4$, $\theta\in (0,1/d)$, there exists a constant $\cstaa$ such that for every $\varepsilon$ satisfying (C), $x\not \in B(0,h_d(\varepsilon))$ we have \begin{align} \label{eq:umajo} u_\varepsilon(x)&\leq \cstb \varphi_d(\varepsilon)^{-1} \val{x}^{2-d},\\ \label{eq:udel} u_\varepsilon(x)&\leq \varphi_d(\varepsilon)^{-1} \val{x}^{2-d}\left[\csta+\cstaa h_d(\varepsilon)^{\theta/2}\right]. \end{align} For $\val{x}>\varepsilon$, we have \begin{align} \label{eq:umino} &u_\varepsilon(x)\geq \csta \varphi_d(\varepsilon)^{-1} \val{x}^{2-d} \quad \text{if $d\geq 5$},\\ \label{eq:umino4} & u_\varepsilon(x)\geq \csta \varphi_4(\varepsilon)^{-1} \val{x}^{-2} \left[1+\log (2\val{x})/\log (1/\varepsilon)\right]^{-1} \quad \text{if $d=4$}. \end{align} We will also often use the following inequality for $\varepsilon$ satisfying (C): $\varphi_d(\varepsilon)h_d(\varepsilon)^d\leq h_d(\varepsilon)^3$. \noindent \textbf{Proof} of theorem \ref{th:rangeX}. Recall that for every $t>0$, $\P^X_\nu$ a.s. the set $\range_t(X)$ is bounded. Thus we only need to consider a bounded Borel set $A$. Thanks to the Markov property of $X$ at time $t$ and theorem \ref{th:tribe} it is clearly enough to prove the second part of theorem \ref{th:rangeX}. Let $\nu\in M_f$ and $\rho<4$ such that $\lim_{\varepsilon\rightarrow 0} \varepsilon^{\rho-d}\val{(\supp \nu)^\varepsilon}=0$. For short we write a.s. for $\P^X_\nu$-a.s. First step. Recall we can write for every $t>0$, $X_t=\sum_{i\in I} Y_t(W^i)$, where $\sum_{i\in I} \delta_{W^i}$ is a Poisson measure on $C(\R^+, \cw)$ with intensity measure $\int \nu(dx) \N_x[\cdot]$. We let $x_0^i$ denote the starting point of the Brownian snake $W^i$ (i.e. $x_0^i= W^i_0(0)$). Notice that a.s. for every $i\in I$, $x^i_0\in \supp \nu$, which is bounded thanks to the hypothesis on $\supp \nu$. Fix $\theta\in (0,1/d)$ such that $d-\rho\geq (d-4)/(1-\theta)$ (and $\theta<4-\rho$ if $d=4$). Fix $R_0$ such that $ \supp \nu\subset B(0, R_0)$. Let $\kappa$ and $\varepsilon_0<1$ fixed as in proposition \ref{th:erange}. We notice that for every bounded Borel set $A\subset B(0, R_0)$, \[ \varphi_d(\varepsilon)\val{\range_0(X)^\varepsilon\cap A}\leq \sum_{i\in I} V_\varepsilon(W^i) + \varphi_d(\varepsilon)\val{A\cap (\supp \nu)^{h_d(\varepsilon)}}, \] where \[ V_\varepsilon(W^i)=\varphi_d(\varepsilon)\val{\range(W^i)^\varepsilon\cap A\cap \bar B(x_0^i,h_d(\varepsilon))^c }. \] We set $V_0(W^i)=\cstap \int_0^\infty ds\;(Y_s(W^i),\ind_A)$. We use the second moment formula for a Poisson measure to get: \begin{multline*} \E^X_\nu\left[\left[\sum_{i\in I} V_\varepsilon(W^i)-\sum_{i\in I} V_0(W^i)\right]^2\right] =\int \nu(dx) \N_x \left[\left[ V_\varepsilon(W)-V_0(W)\right]^2\right]\\ +\left[\int \nu(dx) \N_x \left[ V_\varepsilon(W)-V_0(W)\right]\right]^2. \end{multline*} We deduce from proposition \ref{th:erange} that for every $\varepsilon\in (0,\varepsilon_0]$, \[ \E^X_\nu\left[\left[\sum_{i\in I} V_\varepsilon(W^i)-\sum_{i\in I} V_0(W^i)\right]^2\right] \leq [(\nu,\ind)+(\nu,\ind)^2]h_d(\varepsilon)^\kappa. \] \noindent Notice the hypothesis on $\supp \nu$ and $\theta$ imply that $\lim_{\varepsilon\rightarrow 0} \varphi_d(\varepsilon)\val{ (\supp \nu )^{h_d(\varepsilon)}}=0$. Arguments similar to those used in the first part of the proof of corollary \ref{cor:cvpsrangeY} show then a.s. \[ \lim_{\varepsilon\rightarrow 0} \sum_{i\in I} V_\varepsilon(W^i)=\sum_{i\in I} V_0(W^i). \] Notice we have $\sum_{i\in I} V_0(W^i)=\cstap \int_0^\infty ds \; (X_s,\ind_A)$. Using the above remark on $\supp \nu$, we deduce that a.s. \[ \limsup_{\varepsilon\rightarrow 0} \varphi_d(\varepsilon) \val{\range_0(X)^{\varepsilon} \cap A} \leq \cstap\int_0^\infty ds\;(X_s,\ind_A). \] Second step. To get a lower bound, consider an increasing sequence $(E_p,p\geq 1)$ of measurable subsets of $E=C(\R^+,\cw)$ such that $\bigcup_{p\geq 1} E_p=E$ and $\int \nu(dx) \N_x [E_p]=\alpha_p<\infty $. (For instance we can take $E_p=\left\{W; \sup_{s\geq 0} \zeta_s\geq 1/p\right\}$.) Then a.s. the set $I_p=\left\{i\in I; W^i \in E_p\right\}$ is finite. We have \[ \varphi_d(\varepsilon)\val{\range_0(X)^{\varepsilon} \cap A}\geq \sum_{i\in I_p} V_\varepsilon(W^i) -\sum_{(i,j)\in I_p^2;\; i \neq j} U_\varepsilon(W^i,W^j), \] where \begin{align*} U_\varepsilon(W^i,W^j) &=\varphi_d(\varepsilon) \val{\range(W^i)^{\varepsilon} \cap \range(W^j)^{\varepsilon}\cap A \cap \bar B(x^i_0, h_d(\varepsilon))^c \cap \bar B(x^j_0, h_d(\varepsilon))^c}\\ &=\varphi_d(\varepsilon)\int_{A \cap \bar B(x^i_0, h_d(\varepsilon))^c \cap \bar B(x^j_0, h_d(\varepsilon))^c}dy\; \ind_{\left\{T_{(y,\varepsilon)}(W^i)<\infty \right\}}\ind_{\left\{T_{(y,\varepsilon)}(W^j)<\infty \right\}} . \end{align*} Arguments similar to those of the first step show that a.s. \[ \lim_{\varepsilon\rightarrow 0} \sum_{i\in I_p} V_\varepsilon(W^i) =\sum_{i\in I_p} V_0(W^i)= \sum_{i\in I_p} \cstap\int_0^\infty ds\;(Y_s(W^i),\ind_A). \] Now conditionally on the cardinal of $I_p$, the Brownian snakes $(W^i, i\in I_p)$ are independent and have the same law: $\mu_p=\alpha_p^{-1}\int \nu(dx) \N_x [\cdot\cap E_p]$. For two independent Brownian snakes $(W,W')$ under $\mu_p\otimes \mu_p$, we get using \reff{eq:umajo}, that for $\varepsilon$ satisfying (C), \begin{align*} \mu_p\otimes \mu_p[U_\varepsilon(W,W')] & \leq \alpha_p^{-2} \iint \nu(dx_0) \nu(dx'_0) \N_{x_0}\otimes \N_{x'_0} [U_\varepsilon(W,W')]\\ & \leq \varphi_d(\varepsilon)\alpha_p^{-2} \iint \nu(dx_0) \nu(dx'_0) \int_{A \cap \bar B(x_0, h_d(\varepsilon))^c \cap \bar B(x'_0, h_d(\varepsilon))^c}dy\;\\ &\hspace{4cm}\left[\cstb\varphi_d(\varepsilon)^{-1}\val{y-x_0}^{2-d}\right] \left[\cstb\varphi_d(\varepsilon)^{-1}\val{y-{x'_0}}^{2-d}\right]\\ & \leq \varphi_d(\varepsilon)^{-1}\alpha_p^{-2} (\nu,\ind)^2 \cstb^2 \sup_{x_0\in \R^d} \int_{\bar B(0,R_0) \backslash \bar B(x_0, h_d(\varepsilon))} dy \val{y-x_0}^{4-2d}\\ & \leq \cst \varphi_d(\varepsilon)^{-1} h_d(\varepsilon)^{4-d} \quad \text{if $d\geq 5$}\\ & \leq \cst \varphi_4(\varepsilon)^{-1} \log(\log(1/\varepsilon))\quad \text{if $d=4$}\\ & \leq \cst h_d(\varepsilon)^{\theta/2} \quad \text{if $d\geq 4$}, \end{align*} where the constant $\cst$ is independent of $\varepsilon$ and $A$. Using the Borel-Cantelli lemma for the sequence $(h_d(\varepsilon_n)=n^{-4/\theta},n\geq 1)$, and a monotonicity argument, we get that $\mu_p\otimes \mu_p$-a.s. $\lim_{\varepsilon\rightarrow 0} U_{\varepsilon}(W,W')=0$. Then since the cardinal of $I_p$ is a.s. finite, we get that for every integer $p\geq 1$, a.s., \[ \lim_{\varepsilon\rightarrow 0} \sum_{(i,j)\in I_p^2;\; i \neq j} U_\varepsilon(W^i,W^j)=0. \] We deduce that for every integer $p\geq 1$, a.s. \[ \liminf_{\varepsilon\rightarrow 0} \varphi_d(\varepsilon) \val{\range_0(X)^\varepsilon\cap A} \geq \sum_{i\in I_p} \cstap\int_0^\infty ds\; (Y_s(W^i), \ind_A). \] We get the lower bound by letting $p\rightarrow \infty $. This and the upper bound of the first step ends the proof of the theorem. \findemo \section{Proof of proposition \ref{th:erange}} \label{sec:prprop} We shall use many times in the sequel the fact that $\int_0^\infty ds\; (Y_s,\ind_A)=\int_0^\sigma ds \; \ind_A(\hat W_s)$ $\N_{x_0}$-a.e. We assume $d\geq 4$. We recall easy equalities, which can readily be deduced from the results of section \ref{sec:annexemo}. For every $A\in \cb(\R^d)$, we have \begin{align} \label{eq:N11} \N_x\left[\int_0^\sigma ds\;\ind_A(\hat W_s) \right] &=\int_A dy\; G(x,y),\\ \intertext{where $G$ is the Green kernel in $\R^d$: $G(x,y)=2^{-1}\pi^{-d/2}\Gamma([d-2]/2)\val{x-y}^{2-d}$, and} \label{eq:N12} \N_x\left[\left[\int_0^\sigma ds\;\ind_A(\hat W_s) \right]^2\right] &= 4\int dy\; G(x,y)\left[\int_A dz\; G(y,z)\right]^2. \end{align} We can also compute the first moment under $\ce^*_\rw$. For every $A\in \cb(\R^d)$, $\rw \in \cw$, we have with $\zeta=\zeta_{(\rw)}$, \begin{equation} \label{eq:Egreen} \hspace{-1.5cm} \ce^*_\rw \left[\int_0^\sigma ds\;\ind_A(\hat W_s)\right]= 2\int_0^\zeta dt\; \N_{\rw(t)}\left[\int_0^\sigma ds\;\ind_A(\hat W_s) \right]= 2\int_0^\zeta dt \int_A dy\; G(\rw(t),y). \end{equation} \noindent Thanks to the space invariance of the law of the Brownian snake, we shall only consider the case $x_0=0$ and $A\subset \bar B(0,R_0)$, for $R_0$ fixed. We fix $\theta\in (0,1/d)$ and $R_0>1$. Let $\varepsilon'_0>0$ satisfying (C). We consider $\varepsilon\in (0,\varepsilon'_0)$. In this section, we denote by $\cst$, $c_1,c_2,\ldots$ positive constants whose values depend only on $d,\theta$ and $R_0$. The value of $\cst$ may vary from line to line. For short we shall write $A_\varepsilon=A\cap \bar B(0,h_d(\varepsilon))^c $ (not to be confused with $A^\varepsilon$) and $\range$ for $\range(W)$. We first consider the case $d\geq 5$. Notice that \[ \N_0\left[\val{\range^\varepsilon\cap A_\varepsilon}\right] =\int_{A_\varepsilon} dx \;\N_0\left[T_{(x,\varepsilon)}<\infty \right] =\int_{A_\varepsilon} dx\;u_\varepsilon(x). \] Thus we deduce from \reff{eq:umino} and \reff{eq:udel}, that for $\varepsilon\in (0,\varepsilon'_0)$, \begin{multline*} \csta \varepsilon^{d-4} \int_A dx\; \val{x}^{2-d} - \csta \varepsilon^{d-4} \int_{B(0, \varepsilon^{1-\theta})} dx \; \val{x}^{2-d}\\ %\begin{aligned} \leq \N_0\left[\val{\range^\varepsilon\cap A_\varepsilon}\right] \leq \varepsilon^{d-4}[\csta +\cstaa h_d(\varepsilon)^{\theta/2}] \int_A dx\; \val{x}^{2-d}. %\end{aligned} \end{multline*} Therefore using also \reff{eq:N11}, we have \[ \val{\N_{x_0}\left[\varepsilon^{4-d}\val{\range(W)^\varepsilon\cap A_\varepsilon}-\cstap \int_0^\infty ds\;(Y_s,\ind_A)\right]} \leq \cst h_d(\varepsilon)^{\theta/2}. \] Thus we get the first bound of proposition \ref{th:erange} (take $\kappa<\theta$ and $\varepsilon_0$ small enough). The proof is similar for $d=4$ (use \reff{eq:udel41} instead of \reff{eq:udel1} and the fact that $\val{x}$ is bounded by $R_0$). Now we will prove the second bound. To this end we have to find an upper bound on $I=\N_0\left[\val{\range^\varepsilon\cap A_\varepsilon}^2\right]$ and a lower bound on $J=\N_0\left[\val{\range^\varepsilon\cap A_\varepsilon}\int_0^\sigma ds\; \ind_A(\hat W_s) \right]$. \subsection{An upper bound on $I$} \label{sec:majoI} The term $I$ can also be written \[ I=\iint_{A_\varepsilon\times A_\varepsilon} dx\;dy \; \N_0 \left[ T_{(x,\varepsilon)}<\infty ; T_{(y,\varepsilon)}<\infty \right]. \] Consider the above integral as the sum of the integral over $\val{x-y}\leq 2h_d(\varepsilon)$ (denoted by $I_1$) and the one over $\val{x-y}> 2h_d(\varepsilon)$ (denoted by $I_2$). Using \reff{eq:umajo} we easily obtain an upper bound on $I_1$: \initre \begin{align*}\initer I_1 & \leq \val{B(0,2h_d(\varepsilon))} \int_{A_\varepsilon} dx\; \N_0 \left[ T_{(x,\varepsilon)}<\infty \right]\\ & \leq \cst h_d(\varepsilon)^d \int_{A_\varepsilon} dx\; \varphi_d(\varepsilon)^{-1} \cstb \val{x}^{2-d} \leq \cste \varphi_d(\varepsilon)^{-2}h_d(\varepsilon)^3. \end{align*} \setcounter{cstei}{\thecste} \noindent Notice the event % the set $\left\{T_{(x,\varepsilon)}<\infty ; T_{(y,\varepsilon)}<\infty \right\}$ is a subset of \[ \left\{T_{(x,\varepsilon)}<\infty ; T_{(y,\varepsilon)}\circ \theta_{T_{(x,\varepsilon)}}<\infty \right\}\cup \left\{T_{(y,\varepsilon)}<\infty ; T_{(x,\varepsilon)}\circ \theta_{T_{(y,\varepsilon)}}<\infty \right\}, \] where $\theta_t$ is the usual shift operator. By symmetry, we get \begin{equation} \label{eq:I2majo} \hspace{-2cm}I_2\leq 2\iint_{A_\varepsilon\times A_\varepsilon}dx\;dy\; \ind_{\left\{\val{x-y}>2h_d(\varepsilon)\right\}} \N_0 \left[T_{(x,\varepsilon)}<\infty ; T_{(y,\varepsilon)}\circ \theta_{T_{(x,\varepsilon)}}<\infty\right]. \end{equation} Using the strong Markov property of the Brownian snake under $\N_0$ at the stopping time $T_{(x,\varepsilon)}$ and (\ref{eq:hitting}), we see that the quantity $\N_0\left[T_{(x,\varepsilon)}<\infty ; T_{(y,\varepsilon)}\circ \theta_{T_{(x,\varepsilon)}}<\infty\right]$ is equal to \[ \N_0 \left[T_{(x,\varepsilon)}<\infty ; 2\int_0^{\zeta_{T_{(x,\varepsilon)}}\wedge \tau_{B(y,\varepsilon)}(W_{T(x,\varepsilon)})} dt\;u_\varepsilon\left(W_{T_{(x,\varepsilon)}}(t)-y\right) \expp{\left[-2\int_0^t u_\varepsilon\left(W_{T_{(x,\varepsilon)}}(s)-y\right) ds \right]}\right] . \] Finally the law of the stopped path $W_{T_{(x,\varepsilon)}}$ under $\N_0$ is given by (\ref{eq:Nbeta}). Thus the previous expression is equal to \[ 2\int_0^\infty \!dt\;\E_0 \left[\tau_{B(x,\varepsilon)}>t;\tau_{B(y,\varepsilon)}>t; u_\varepsilon(\beta_t-x)u_\varepsilon(\beta_t-y) \expp{\left[-2\int_0^t ds\; \left[u_\varepsilon(\beta_s-x)+u_\varepsilon(\beta_s-y) \right]\right]} \right]. \] We substitute this last expression for $\N_0\left[T_{(x,\varepsilon)}<\infty ; T_{(y,\varepsilon)}\circ \theta_{T_{(x,\varepsilon)}}<\infty\right]$ in (\ref{eq:I2majo}), and then decompose the right-hand side of (\ref{eq:I2majo}) in three terms by considering the integral in $dxdy$ on the sets $\val{\beta_t-x}\wedge\val{\beta_t-y}>h_d(\varepsilon)$ (integral $I_{21}$), $\val{\beta_t-x}\leq h_d(\varepsilon)$ (integral $I_{22}$), and $\val{\beta_t-y}\leq h_d(\varepsilon)$ (integral $I_{23}$) (recall $\val{x-y}>2h_d(\varepsilon)$). \textbf{An upper bound on $I_{21}$.} We shall need the following notation: \[ I_0=4 \csta^2 \int dz \; G(0,z) \left[\int_{A}dx\; \val{z-x}^{2-d}\right]^2. \] We use \reff{eq:udel} to bound $I_{21}$ above by: for $\varepsilon\in (0,\varepsilon'_0)$, \initre \begin{align*}\initer &4\iint_{A_\varepsilon\times A_\varepsilon}dx\;dy\; \ind_{\{\val{x-y}>2h_d(\varepsilon)\}} \int_0^\infty dt\; \E_0\Big[ \val{\beta_t-x}>h_d(\varepsilon); \val{\beta_t-y}>h_d(\varepsilon) ;\\ &\hspace{1cm}\varphi_d(\varepsilon)^{-2}\val{ \beta_t-x }^{2-d} \val{\beta_t-y}^{2-d} \left(\csta +\cstaa h_d(\varepsilon)^{\theta/2}\right)^2 \Big] \\ &\hspace{.5cm} \leq 4 \varphi_d(\varepsilon)^{-2} \left[\csta^2 +\cst h_d(\varepsilon)^{\theta/2}\right] \iint_{A\times A}dx\;dy\; \int dz \; G(0,z) \val{z-x}^{2-d}\val{z-y}^{2-d}\\ &\hspace{.5cm} \leq \varphi_d(\varepsilon)^{-2} I_0+\cste \varphi_d(\varepsilon)^{-2}h_d(\varepsilon)^{\theta/2}. \end{align*} \setcounter{csteii}{\thecste} \textbf{An upper bound on $I_{22}$ and $I_{23}$.} By symmetry we have $I_{22}=I_{23}$. Before getting an upper bound on $I_{22}$, notice that $\val{\beta_t-x}\leq h_d(\varepsilon)$ and $\val{x-y}>2h_d(\varepsilon)$ imply $\val{\beta_t-y}>h_d(\varepsilon)$. Furthermore thanks to \reff{eq:umajo}, we get \initre \begin{align*}\initer \int_{A_\varepsilon} dy\;\ind_{\left\{\val{\beta_t-y}>h_d(\varepsilon)\right\}} u_\varepsilon(\beta_t-y) \expp{-2\int_0^t u_\varepsilon(\beta_s-y) ds} & \leq \int_{A} dy\;\left[\cstb \varphi_d(\varepsilon)^{-1}\val{\beta_t-y}^{2-d} \right] \\ &\leq \cstb \varphi_d(\varepsilon)^{-1} \int_{B(0,R_0)} dy\; \val{y}^{2-d} =\cste \varphi_d(\varepsilon)^{-1}. \end{align*} Thus the sum $I_{22}+I_{23}$ is bounded above by \[ 8\mcste \varphi_d(\varepsilon)^{-1}\int_{A_\varepsilon}dx\; \int_0^\infty dt\; \E_0 \left[\tau_{B(x,\varepsilon)}>t; \ind_{\left\{\val{\beta_t-x}\leq h_d(\varepsilon)\right\}} u_\varepsilon(\beta_t-x) \expp{-2\int_0^t u_\varepsilon(\beta_s-x)ds } \right]. \] Using the Cauchy-Schwarz inequality and formula \reff{eq:fk}, we get \initre \begin{align*}\initer I_{22}+I_{23} &\leq 8\mcste \varphi_d(\varepsilon)^{-1} \left[\int_{A_\varepsilon}dx\; \int_0^\infty dt \;\P_0 \left[ {\val{\beta_t-x}\leq h_d(\varepsilon)}\right]\right]^{1/2} \\ &\phantom{\leq 2\mcste R_0^2 \varepsilon^{d-4} } \times \left[\int_{A_\varepsilon}dx\; \int_0^\infty dt\; \E_0 \left[ \tau_{B(x,\varepsilon)}>t; u_\varepsilon(\beta_t-x)^2 \expp{-4\int_0^t u_\varepsilon(\beta_s-x)ds }\right]\right]^{1/2} \\ &\leq 8\mcste \varphi_d(\varepsilon)^{-1} \left[\int_{A}dx\; \int dz \;G(0,z) \ind_{ \left\{\val{z-x}\leq h_d(\varepsilon)\right\}}\right]^{1/2} \left[\int_{A_\varepsilon}dx\; 2^{-1} u_\varepsilon(x) \right]^{1/2}. \end{align*} Then thanks to \reff{eq:umajo}, we get $I_{22}+I_{23} \leq \cste \varphi_d(\varepsilon)^{-3/2}h_d(\varepsilon)^{d/2}\leq \mcste \varphi_d(\varepsilon)^{-2}h_d(\varepsilon)^{3/2} $. \setcounter{csteiii}{\thecste} \textbf{Conclusion on the upper bound on $I$.} By combining the previous results, we get for $d\geq 4$ \[ I\leq \cstei \varphi_d(\varepsilon)^{-2}h_d(\varepsilon)^{3} +\varphi_d(\varepsilon)^{-2} I_0 +\csteii \varphi_d(\varepsilon)^{-2}h_d(\varepsilon)^{\theta/2} +\csteiii \varphi_d(\varepsilon)^{-2}h_d(\varepsilon)^{3/2}. \] Thus we get $ \varphi_d(\varepsilon)^2 I\leq I_0+\cste h_d(\varepsilon)^{\theta/2}$. \setcounter{csteiii}{\thecste} \subsection{A lower bound on $J$} \label{sec:sublbJ} We shall need the last hitting time of $\bar B(x,\varepsilon)$ under $\N_0$ for the Brownian snake: \[ L_{(x,\varepsilon)}=\sup \left\{s\geq 0; \exists t\in [0,\zeta_s], W_s(t)\in \bar B(x,\varepsilon)\right\}. \] We then get \initre \begin{multline*}\initer J=\int_{A_\varepsilon} dx \;\N_0 \left[T_{(x,\varepsilon)}<\infty ; \int_0^{L_{(x,\varepsilon)}} ds \;\ind_A(\hat W_s) \right]\\ +\int_{A_\varepsilon} dx \;\N_0 \left[T_{(x,\varepsilon)}<\infty ; \int_{T_{(x,\varepsilon)}}^\sigma ds \;\ind_A(\hat W_s) \right]\\ -\int_{A_\varepsilon} dx \;\N_0 \left[T_{(x,\varepsilon)}<\infty ; \int_{T_{(x,\varepsilon)}}^{L_{(x,\varepsilon)}} ds \;\ind_A(\hat W_s) \right] . \end{multline*} The time-reversal invariance property of the It\^o measure and the characterization of the excursion measure $\N_x$ readily imply that the latter itself enjoys the same invariance property. Thus the first two terms of the right-hand side are equal. We shall denote their sum by $J_1$. Let $J_2$ denote the third term. \textbf{A lower bound on $J_1$.} Let us use the strong Markov property of the Brownian snake at time $T_{(x,\varepsilon)}$, then (\ref{eq:Egreen}) and (\ref{eq:Nbeta}), to get \initre \begin{align*}\initer J_1 &= 2 \int_{A_\varepsilon} dx \;\N_0 \left[T_{(x,\varepsilon)}<\infty ; 2 \int_0^{\zeta_{T_{(x,\varepsilon)}}} dt \; \int_A dy \; G\left(W_{T_{(x,\varepsilon)}} (t),y\right) \right]\\ &= 4 \int_{A_\varepsilon} dx \int_A dy \int_0^\infty dt \;\E_0 \left[\tau_{B(x,\varepsilon)}> t ; G(\beta_t,y) u_\varepsilon(\beta_t-x)\expp{-2\int_0^t u_\varepsilon(\beta_s-x)ds}\right]. \end{align*} Fatou's lemma gives that $ \liminf_{\varepsilon\rightarrow 0} \varphi_d(\varepsilon) J_1\geq J_0$, where \[ J_0=4\csta \iint_{A\times A} dxdy\int dz \; G(0,z)G(z,y) \val{z-x}^{2-d}. \] Unfortunately, we need an estimate on the rate of convergence. This requires some technical calculations. Notice that on $\{\tau_{B(x,h_d(\varepsilon))}(\beta)>t\}$, inequalities \reff{eq:umino}, \reff{eq:umino4} and \reff{eq:umajo} imply \[ \csta \varphi_d(\varepsilon)^{-1} F_d(\beta_t-x) \val{\beta_t-x}^{2-d}\leq u_\varepsilon(\beta_t -x)\leq\cstb \varphi_d(\varepsilon)^{-1} \val{\beta_t-x}^{2-d}, \] where $F_d(z)=1$ if $d\geq 5$ and $F_4(z)=\left[1+\log (2\val{z})/\log(1/\varepsilon)\right]^{-1}$. For short we write $\Gamma_t=2\cstb \varphi_d(\varepsilon)^{-1} \int_0^t \val{\beta_s-x}^{2-d} ds$. Then $\varphi_d(\varepsilon)J_1$ is bounded below by \[ J'_1=4\csta \int_{A_\varepsilon}\!dx \int_A \!dy \int_0^\infty \!dt \;\E_0\left[\tau_{B(x,h_d(\varepsilon))}>t; G(\beta_t,y) \val{\beta_t-x}^{2-d}F_d(\beta_t-x)\expp{-\Gamma_t }\right]. \] In order to obtain an upper bound on $\val{ J'_1-J_0}$, we have to find an upper bound on \[ \iint_{A\times A}dx dy \int_0^\infty dt \;\E_0\left[ G(\beta_t,y)\val{\beta_t-x}^{2-d} \left[1-\ind_{A_\varepsilon}(x) \ind_{ \left\{\tau_{B(x,h_d(\varepsilon))}>t\right\}} F_d(\beta_t-x)\expp{-\Gamma_t }\right]\right]. \] Thus we shall decompose $1-\ind_{A_\varepsilon}(x) \ind_{\left\{\displaystyle \tau_{B(x,h_d(\varepsilon))}>t\right\}} F_d(\beta_t-x) \expp{-\Gamma_t }$ into a sum of four terms: \begin{multline*} \left[1-\ind_{A_\varepsilon}(x)\right] +\ind_{A_\varepsilon}(x)\left[1-\ind_{\left\{ \tau_{B(x,h_d(\varepsilon))}>t\right\}} \right] \\ +\ind_{A_\varepsilon}(x)\ind_{\left\{ \tau_{B(x,h_d(\varepsilon))}>t\right\}}\left[1-F_d(\beta_t-x) \right] +\ind_{A_\varepsilon}(x)\ind_{\left\{ \tau_{B(x,h_d(\varepsilon))}>t\right\}}F_d(\beta_t-x)\left[1-\expp{-\Gamma_t } \right] . \end{multline*} We denote by $J_{11}$, $J_{12}$, $J_{13}$ and $J_{14}$ the corresponding integrals. The integral \[ J_{11}= \int_{A\backslash A_\varepsilon}dx \int_A dy \int_0^\infty dt \;\E_0\left[ G(\beta_t,y)\val{\beta_t-x}^{2-d} \right] \] is easily bounded above by \[ \int_{B(0,h_d(\varepsilon))} dx\; \int_{B(0,R_0)} dy \int dz \; G(0,z) G(z,y)\val{z-x}^{2-d} \leq \cste h_d(\varepsilon)^{2}. \] We bound $J_{12}$ by applying the strong Markov property of Brownian motion at time $\tau_{B(x,h_d(\varepsilon))}$, \initre\setcounter{csteij}{\thecste} \begin{align*}\initer J_{12}&=\int_{A_\varepsilon}dx \int_A dy \int_0^\infty dt \;\E_0\left[\tau_{B(x,h_d(\varepsilon))}\leq t ; G(\beta_t,y) \val{\beta_t-x}^{2-d}\right]\\ %&\hspace{1cm} &\leq \int_{A_\varepsilon}dx \int_A dy\; \E_0\left[\tau_{B(x,h_d(\varepsilon))}<\infty ; \int dz \;G(\beta_{\tau_{B(x,h_d(\varepsilon))}},z) G(z,y) \val{z-x}^{2-d}\right]. \end{align*} An easy calculation shows that there exists a constant $\cste$ such that for every $(x,x')\in B(0,2R_0)\times B(0,2R_0)$, $\val{x-x'}\leq 1/2$, \[ \int_{B(0,R_0)}dy \int dz \;G(x',z)G(z,y) \val{z-x}^{2-d} \leq \mcste \varphi_d(\val{x'-x}) \] Furthermore %thanks to proposition 1.6 p56 from \cite{ps:bm}, we get for we have for every $r\in (0,1)$, \begin{equation} \label{eq:intta} \int_{B(0,R_0)}dx \;\P_0\left[\tau_{B(x,r)}<\infty \right] =\int_{B(0,R_0)}dx\; \left[\left(\frac{r}{\val{x}}\right)^{d-2}\wedge 1\right] \leq \cst r^{d-2}. \end{equation} We deduce from the previous remarks that if $d\geq 5$, \initre \[ \initer J_{12} \leq \cst h_d(\varepsilon)^{4-d} \int_{A_\varepsilon} dx\; \P_0\left[ \tau_{B(x,h_d(\varepsilon))}<\infty \right] \leq \cst h_d(\varepsilon)^{4-d+d-2}=\cst h_d(\varepsilon)^2, \] and if $d=4$, $J_{12}\leq \cst \log(1/h_d(\varepsilon)) h_d(\varepsilon)^2 $. Thus we get that for $d\geq 4$, $J_{12}\leq \cste h_d(\varepsilon)^{3/2}$. \setcounter{cstej}{\thecste} \noindent If $d\geq 5$ then $J_{13}=0$. For $d=4$ thanks to \reff{eq:lolo} we have for $\val{z}\geq h_4(\varepsilon)$, $\val{1-F_4(z)}\leq 2 \val{\log(2\val{z})}/\log(1/\varepsilon)$. We deduce that \initre \begin{align*}\initer J_{13} &\leq \log(1/\varepsilon)^{-1} \iint_{A\times A}\!\!dx dy \int_0^\infty \!dt\; \E_0\left[\tau_{B(x,h_d(\varepsilon))}>t; G(\beta_t,y)2\val{\log({2\val{\beta_t-x}})} \val{\beta_t-x}^{-2}\right]\\ &\leq \cst \log(1/\varepsilon)^{-1} \iint_{A\times A}dx dy \int dz\; G(0,z)\val{\log({2\val{z-x}})} \val{z-x}^{-2}G(z,y)\\ &\leq \cst \log(1/\varepsilon)^{-1} \leq \cste h_d(\varepsilon)^\theta. \end{align*} \noindent Notice first that thanks to \reff{eq:lolo}, $F_d(z)\leq 2$ for $\val{z}\geq h_d(\varepsilon)$. We have, using the Markov property for Brownian motion at time $s$, \setcounter{cstejjiii}{\thecste} \initre \begin{align*}\initer J_{14} &\leq 2\iint_{A\times A}dx dy \int_0^\infty dt\; \\ &\hspace{2cm}\E_0\Bigg[ \tau_{B(x,h_d(\varepsilon))}>t; G(\beta_t,y) \val{\beta_t-x}^{2-d}2\cstb \varphi_d(\varepsilon)^{-1}\int_0^t \val{\beta_s-x}^{2-d} ds\Bigg]\\ &\leq \cst\varphi_d(\varepsilon)^{-1}\iint_{A\times A} dxdy\;\int_0^\infty ds\; \int_0^\infty dt\;\\ &\hspace{2cm} \E_{0}\left[\val{\beta_s-x}^{2-d}\E_{\beta_s} \left[\val{\beta_{t}-x}>h_d(\varepsilon) ; G(\beta_t,y)\val{\beta_{t}-x}^{2-d}\right]\right]\\ &\leq \cst \varphi_d(\varepsilon)^{-1} M(d,h_d(\varepsilon)), \end{align*} where \[ M(d,\varepsilon)=\iint_{B(0,R_0)^2} dxdy\;\iint dzdz'\; G(0,z)\val{z-x}^{2-d} G(z,z') G(z',y)\val{z'-x}^{2-d}\ind_{\val{z'-x}>\varepsilon}. \] An easy computation shows there exists a constant $c$ such that for $\varepsilon\in (0,1]$, \begin{equation} \label{eq:Mde} M(d,\varepsilon)\leq \left\{ \begin{array}{ll} \cst & \text{ if } d\in \{4,5\},\\ \cst+ \cst \log(1/\varepsilon) & \text{ if } d=6,\\ \cst \varepsilon^{6-d} & \text{ if } d\geq 7. \end{array} \right. \end{equation} Thus we easily deduce that $J_{14}\leq \cste h_d(\varepsilon)^\theta$. We have $\varepsilon^{4-d}J_1\geq J_0-4\csta (J_{11}+J_{12}+J_{13}+J_{14})$. Putting together the previous results, we get for $d\geq 4$, \setcounter{csteji}{\thecste} \[ \varepsilon^{4-d}J_1\geq J_0- 4\csta [\csteij h_d(\varepsilon)^2+\cstej h_d(\varepsilon)^{3/2} +\cstejjiii h_d(\varepsilon)^\theta +\csteji h_d(\varepsilon)^\theta]\geq J_0- \cste h_d(\varepsilon)^\theta. \] \setcounter{cstejii}{\thecste} \textbf{An upper bound on $J_2$.} We will first recall the decomposition of the Brownian snake under $\ce^*_{\rw}$ (see theorem 2.5 in \cite{lg:pvmppde}). We denote by $(\alpha_i,\beta_i)$, $i\in I$, the excursion intervals of $\zeta$ above its minimum process (i.e. of the process $(\zeta_t-\inf_{s\in [0,t]}\zeta_s)$ above $0$) before $\sigma$ under $\ce^*_{\rw}$. For $i\in I$ the paths $W_s, s\in [\alpha_i,\beta_i]$ coincide over $[0,\zeta_{\alpha_i}]$. For every $i\in I$, and $s\geq 0$ we set $W_s^i(t)=W_{(\alpha_i+s)\wedge \beta_i}(t+\zeta_{\alpha_i})$, $t\in [0,\zeta^i_s]$ with $\zeta^i_s=\zeta_{(\alpha_i+s)\wedge \beta_i}-\zeta_{\alpha_i}$. Then $W^i_s$ is a stopped path ($W^i_s\in \cw$) with initial point $W_{(\alpha_i+s)\wedge \beta_i}(\zeta_{\alpha_i})=\hat W_{\alpha_i}=\rw(\zeta_{\alpha_i})$. \begin{proposition}[Le Gall] The random measure $\sum_{i\in I} \delta_{(\zeta_{\alpha_i},W^i)}$ is under $\ce^*_{\rw}$ a Poisson point measure on $[0,\zeta_{(\rw)}]\times C(\R^+,\cw)$ with intensity $2 dt\; \N_{\rw(t)}[\cdot]$. \end{proposition} The process $\sum_{i\in I, \zeta_{\alpha_i}\leq t} \delta_{W^i}$ for $t\in [0,\zeta_{\rw}]$ is a Poisson point process with inhomogeneous intensity. We will now describe the law under $\ce^*_{W_{T_{(x,\varepsilon)}}}$ of the first excursion $(\zeta_{\alpha_{i_0}}, W^{i_0})$ which hits the ball $\bar B(x,\varepsilon)$, that is, with evident notation, the first excursion for which $T_{(x,\varepsilon)}(W^i)$ is finite. We first notice that under $\N_0[\cdot\mid T_{(x,\varepsilon)}<\infty ]$, $\ce^*_{W_{T_{(x,\varepsilon)}}}$-a.s. there are such excursions. Indeed we have thanks to lemma 2.1 of \cite{lg:hppt} that $\N_0[.\mid T_{(x,\varepsilon)}<\infty ]$-a.s. %\begin{align*} \[ \ce^*_{W_{T_{(x,\varepsilon)}}}[\exists i\in I, T_{(x,\varepsilon)}(W^i)<\infty ] %&=1-\exp{-2 \int_0^{\zeta_{T_{(x,\varepsilon)}}} dt\; \N_{W_{T_{(x,\varepsilon)}}(t)}[T_{(x,\varepsilon)}<\infty ]}\\ =1-\exp{-2 \int_0^{\zeta_{T_{(x,\varepsilon)}}} dt\; u_\varepsilon(W_{T_{(x,\varepsilon)}}(t)-x)}=1. \] %\end{align*} Since the integral $\int_0^r dt\; u_\varepsilon(W_{T_{(x,\varepsilon)}}(t)-x)$ is finite for $r<\zeta_{T_{(x,\varepsilon)}}$, we deduce there exists a unique first excursion $i_0$ which hits $\bar B(x,\varepsilon)$. Classical arguments on Poisson point process implies that the law of $(\zeta_{\alpha_{i_0}}, W^{i_0})$ is $2 \ind_{[0,\zeta_{{T_{(x,\varepsilon)}}})}(t)dt\; \N_{W_{T_{(x,\varepsilon)}}(t)}[T_{(x,\varepsilon)}<\infty ,\cdot]$. We introduce the random time $M_{(x,\varepsilon)}=\inf \left\{s>T_{(x,\varepsilon)}; \zeta_s=m(T_{(x,\varepsilon)}, L_{(x,\varepsilon)})\right\}$. It is clear from the definition of the excursion $i_0$ that ${\alpha_{i_0}}=M_{(x,\varepsilon)}$ under $\ce^*_{W_{T_{(x,\varepsilon)}}}$. We will now express $J_2$ using the excursion $i_0$. We have \begin{align*} J_2 &=2\int_{A_\varepsilon} dx\; \N_0\left[T_{(x,\varepsilon)}<\infty ; \int_{M_{(x,\varepsilon)}}^{L_{(x,\varepsilon)}}ds\;\ind_A (\hat W_s)\right]\\ &=2\int_{A_\varepsilon} dx\; \N_0\left[T_{(x,\varepsilon)}<\infty ; \ce^*_{W_{T_{(x,\varepsilon)}}}\left[\int_{M_{(x,\varepsilon)}}^{L_{(x,\varepsilon)}}ds\;\ind_A (\hat W_s)\right]\right]\\ &=2\int_{A_\varepsilon} dx\; \N_0\left[T_{(x,\varepsilon)}<\infty ; \ce^*_{W_{T_{(x,\varepsilon)}}}\left[\int_{\alpha_{i_0}}^{L_{(x,\varepsilon)}(W^{i_0})}ds\;\ind_A (\hat W_s^{i_0})\right]\right]\\ &=4\int_{A_\varepsilon} dx\; \N_0\left[T_{(x,\varepsilon)}<\infty ; \int_0^{\zeta_{{T_{(x,\varepsilon)}}}}dt\; \N_{W_{T_{(x,\varepsilon)}}(t)}\left[T_{(x,\varepsilon)}<\infty ; \int_{0}^{L_{(x,\varepsilon)}}ds\;\ind_A (\hat W_s)\right]\right]. \end{align*} We used the time reversal property of the Brownian snake for the first equality, then the strong Markov property and at last the definition of the excursion $i_0$ and its law. We will distinguish according to $\left\{t\geq \tau_{B(x,h_d(\varepsilon))}\right\}$ (integral $J_{21}$) and $\left\{t<\tau_{B(x,h_d(\varepsilon))}\right\}$ (integral $J_{22}$). Notice that since $x\in A_\varepsilon$ we have $\tau_{B(x,h_d(\varepsilon))}(W_{T_{(x,\varepsilon)}})<\zeta_{T_{(x,\varepsilon)}}$ $\N_0$-a.e. \noindent We now bound $J_{21}$ using \reff{eq:N11}. \begin{align*} J_{21} &= 4\int_{A_\varepsilon} dx\; \N_0\left[T_{(x,\varepsilon)}<\infty ; \int_{\tau_{B(x,h_d(\varepsilon))}}^{\zeta_{{T_{(x,\varepsilon)}}}}dt\; \N_{W_{T_{(x,\varepsilon)}}(t)}\left[T_{(x,\varepsilon)}<\infty ; \int_{0}^{L_{(x,\varepsilon)}}ds\;\ind_A (\hat W_s)\right]\right]\\ &\leq 4\int_{A_\varepsilon} dx\; \N_0\left[T_{(x,\varepsilon)}<\infty ; \int_{\tau_{B(x,h_d(\varepsilon))}}^{\zeta_{{T_{(x,\varepsilon)}}}}dt\; \N_{W_{T_{(x,\varepsilon)}}(t)}\left[\int_{0}^{\sigma}ds\;\ind_A (\hat W_s)\right]\right]\\ &=4 \int_{A_\varepsilon} dx\; \N_0\left[T_{(x,\varepsilon)}<\infty ; \int_{\tau_{B(x,h_d(\varepsilon))}}^{\zeta_{{T_{(x,\varepsilon)}}}}dt\; \int_A dy\; G({W_{T_{(x,\varepsilon)}}}(t),y)\right]. \end{align*} Now we use \reff{eq:Nbeta}, the Cauchy-Schwarz inequality and \reff{eq:fk} to get \begin{align*} J_{21} &\leq 4 \int_{A_\varepsilon} dx\int_0^\infty dt \; \E_0\left[\tau_{B(x,\varepsilon)}>t\geq \tau_{B(x,h_d(\varepsilon))}; \int_A dy\; G(\beta_t,y)u_\varepsilon(\beta_t-x)\expp{-2\int_0^t u_\varepsilon(\beta_r-x)dr}\right]\\ &\leq 4 \left[2^{-1}\int_{A_\varepsilon} dx\;u_\varepsilon(x) \right]^{1/2} \left[\int_{A_\varepsilon} dx\int_0^\infty dt \; \E_0\left[t\geq \tau_{B(x,h_d(\varepsilon))}; \left(\int_A dy\; G(\beta_t,y)\right)^2\right]\right]^{1/2}\\ &\leq \cst \varphi_d(\varepsilon)^{-1/2} \left[\int_{A_\varepsilon} dx\;\P_0\left[\tau_{B(x,h_d(\varepsilon))}<\infty\right ] \sup_{x'\in B(0,2R_0)} \int dz\; G(z,x') \left(\int_A dy\; G(z,y)\right)^2\right]^{1/2}\\ &\leq \cst \varphi_d(\varepsilon)^{-1/2} h_d(\varepsilon)^{(d-2)/2}. \end{align*} We used the strong Markov property at time $\tau_{B(x,h_d(\varepsilon))}$ and \reff{eq:intta} for the last two inequalities. This implies that $J_{21}\leq \cste \varphi_d(\varepsilon)^{-1}h_d(\varepsilon)^{1/2}$. \setcounter{cstejji}{\thecste} Using the time reversal property of the Brownian snake, the strong Markov property at time $T_{(x,\varepsilon)}$ and \reff{eq:Egreen} we get \begin{align*} J_{22} &=4\int_{A_\varepsilon} dx\; \N_0\Bigg[T_{(x,\varepsilon)}<\infty ; \int_0^{\tau_{B(x,h_d(\varepsilon))}}dt\; \N_{W_{T_{(x,\varepsilon)}}(t)}\left[T_{(x,\varepsilon)}<\infty ; \int_{T_{(x,\varepsilon)}}^{\sigma}ds\;\ind_A (\hat W_s)\right]\Bigg]\\ %&=4\int_{A_\varepsilon} dx\; \N_0\Bigg[T_{(x,\varepsilon)}<\infty ; \int_0^{\tau_{B(x,h_d(\varepsilon))}}dt\; \\ %&\hspace{3cm} \N_{W_{T_{(x,\varepsilon)}}(t)}\left[T_{(x,\varepsilon)}<\infty ; \ce^*_{W_{T_{(x,\varepsilon)}}} \left[\int_{0}^{\sigma}ds\;\ind_A (\hat W_s)\right]\right]\Bigg]\\ &=8\int_{A_\varepsilon} dx\; \N_0\Bigg[T_{(x,\varepsilon)}<\infty ; \int_0^{\tau_{B(x,h_d(\varepsilon))}}dt\; \\ &\hspace{3cm}\N_{W_{T_{(x,\varepsilon)}}(t)}\left[T_{(x,\varepsilon)}<\infty ; \int_0^{\zeta_{T_{(x,\varepsilon)}}}ds\int_{A} dy\; G(W_{T_{(x,\varepsilon)}}(s),y)\right]\Bigg]. \end{align*} We will distinguish according to $\left\{s\geq \tau_{B(x,h_d(\varepsilon))}\right\}$ (integral $J_{23}$) and $\left\{s< \tau_{B(x,h_d(\varepsilon))}\right\}$ (integral $J_{24}$). We now bound $J_{23}$. We have \begin{align*} J_{23} &=8\int_{A_\varepsilon} dx\; \N_0\Bigg[T_{(x,\varepsilon)}<\infty ; \int_0^{\tau_{B(x,h_d(\varepsilon))}}dt\; \\ &\hspace{2cm}\N_{W_{T_{(x,\varepsilon)}}(t)}\left[T_{(x,\varepsilon)}<\infty ; \int_{\tau_{B(x,h_d(\varepsilon))}}^{\zeta_{T_{(x,\varepsilon)}}}ds\int_{A} dy\; G(W_{T_{(x,\varepsilon)}}(s),y)\right]\Bigg]\\ &=8\int_{A_\varepsilon} dx \int_0^\infty dt\;\E_0\Bigg[\tau_{B(x,h_d(\varepsilon))}> t; u_\varepsilon(\beta_t-x)\expp{-2\int_0^t u_\varepsilon(\beta_r-x)dr}\int_0^\infty ds\;\\ &\hspace{2cm} \E_{\beta_t}\left[\tau_{B(x,\varepsilon)}>s\geq \tau_{B(x,h_d(\varepsilon))}; \int_{A} dy\; G(\beta_s,y) u_\varepsilon(\beta_s-x)\expp{-2\int_0^s u_\varepsilon(\beta_v-x)dv}\right]\Bigg]\\ &\leq \cst \varphi_d(\varepsilon)^{-1}\int_{A_\varepsilon} dx \int_0^\infty dt\;\E_0\Bigg[\tau_{B(x,h_d(\varepsilon))}>t; \val{\beta_t-x}^{2-d}\\ &\hspace{2cm}\left[\int_0^\infty ds\; \E_{\beta_t}\left[\tau_{B(x,\varepsilon)}>s; u_\varepsilon(\beta_s-x)^2\expp{-4\int_0^s u_\varepsilon(\beta_v-x)dv}\right]\right]^{1/2}\\ &\hspace{2cm}\left[\int_0^\infty ds\; \E_{\beta_t}\left[s\geq \tau_{B(x,h_d(\varepsilon))}; \left(\int_{A} dy\; G(\beta_s,y)\right)^2\right]\right]^{1/2}\Bigg]\\ &\leq \cst \varphi_d(\varepsilon)^{-1}\int_{A_\varepsilon} dx\; \int_0^\infty dt\;\E_0\Bigg[\tau_{B(x,h_d(\varepsilon))}> t; \val{\beta_t-x}^{2-d}\left[2^{-1}u_\varepsilon(\beta_t-x)\right]^{1/2}\\ &\hspace{2cm}\left[\E_{\beta_t}\left[\tau_{B(x,h_d(\varepsilon))}<\infty ; \E_{\beta_{\tau_{B(x,h_d(\varepsilon))}}}\left[\int_0^\infty ds\; \left(\int_{A} dy\; G(\beta_s,y)\right)^2\right]\right]\right]^{1/2}\Bigg]\\ &\leq \cst \varphi_d(\varepsilon)^{-3/2}\int_{A_\varepsilon} dx\; \int_0^\infty dt\;\E_0\left[\tau_{B(x,h_d(\varepsilon))}> t; \val{\beta_t-x}^{(6-3d)/2} \P_{\beta_t}\left[\tau_{B(x,h_d(\varepsilon))}<\infty \right]^{1/2}\right]\\ &\hspace{2cm}\left[\sup_{x'\in B(0,2R_0)}\int dz'G(x',z')\left(\int_{A} dy\; G(z',y)\right)^2\right]^{1/2}\\ &\leq \cst \varphi_d(\varepsilon)^{-3/2}\int_{A_\varepsilon} dx\; \int_{\val{z-x}\geq h_d(\varepsilon)} dz\; G(0,z)\val{z-x}^{(6-3d)/2}h_d(\varepsilon)^{(d-2)/2}\val{z-x}^{(2-d)/2}. \end{align*} We used \reff{eq:Nbeta} twice for the second equality, \reff{eq:umajo} and Cauchy-Schwarz inequality for the first inequality, \reff{eq:fk} and the strong Markov property at time $\tau_{B(x,h_d(\varepsilon))}$ for the second and \reff{eq:intta} for the last. We easily deduce that $J_{23}\leq \cste \varphi_d(\varepsilon)^{-1}h_d(\varepsilon)$. \setcounter{cstejjii}{\thecste} \noindent For $J_{24}$ we have using \reff{eq:Nbeta} twice and \reff{eq:umajo} twice, \begin{align*} J_{24} &=8\int_{A_\varepsilon} dx\; \N_0\Bigg[T_{(x,\varepsilon)}<\infty ; \int_0^{\tau_{B(x,h_d(\varepsilon))}}dt\; \\ &\hspace{4cm}\N_{W_{T_{(x,\varepsilon)}}(t)}\left[T_{(x,\varepsilon)}<\infty ; \int_0^{\tau_{B(x,h_d(\varepsilon))}}ds\int_{A} dy\; G(W_{T_{(x,\varepsilon)}}(s),y)\right]\Bigg]\\ &=8\int_{A_\varepsilon} dx\; \int_0^\infty dt\int_0^\infty ds\; \E_0\Bigg[\tau_{B(x,h_d(\varepsilon))}> t; u_\varepsilon(\beta_t-x)\expp{-2\int_0^t u_\varepsilon(\beta_r-x)dr}\\ &\hspace{4cm}\E_{\beta_t}\left[\tau_{B(x,h_d(\varepsilon))}>s; \int_{A} dy\; G(\beta_s,y) u_\varepsilon(\beta_s-x)\expp{-2\int_0^s u_\varepsilon(\beta_v-x)dv}\right]\Bigg]\\ &\leq \cst \varphi_d(\varepsilon)^{-2} \int_{A_\varepsilon} dx\; \int_0^\infty dt\int_0^\infty ds\; \E_0\Bigg[\tau_{B(x,h_d(\varepsilon))}> t; \val{\beta_t-x}^{2-d}\\ &\hspace{4cm}\E_{\beta_t}\left[\tau_{B(x,h_d(\varepsilon))}>s; \int_{A} dy\; G(\beta_s,y)\val{\beta_s-x}^{2-d}\right]\Bigg]\\ &\leq \cst \varphi_d(\varepsilon)^{-2} M(d,h_d(\varepsilon)). \end{align*} Using \reff{eq:Mde} we get $J_{24}\leq \cste \varphi_d(\varepsilon)^{-1}h_d(\varepsilon)^\theta$. As a conclusion we get \[ J_2\leq \cstejji \varphi_d(\varepsilon)^{-1}h_d(\varepsilon)^{1/2}+\cstejjii \varphi_d(\varepsilon)^{-1}h_d(\varepsilon)+\mcste \varphi_d(\varepsilon)^{-1}h_d(\varepsilon)^{\theta}. \] \textbf{Conclusion on the lower bound on $J$.} By combining the previous results, we get for $d\geq 4$, \[ \varphi_d(\varepsilon)J\geq J_0- \cstejii h_d(\varepsilon)^{\theta} -\varphi_d(\varepsilon) J_2\geq J_0- \cste h_d(\varepsilon)^{\theta}. \] \subsection{End of the proof of proposition \ref{th:erange}} We deduce from formula (\ref{eq:N12}), that \[ J_0= \cstap \N_0\left[\left[\int_0^\sigma \ind_A(\hat W_s) ds \right]^2\right],\quad \text{and} \quad I_0= {\cstap}^2 \N_0\left[\left[\int_0^\sigma \ind_A(\hat W_s) ds \right]^2\right]. \] Thus we get from section \ref{sec:majoI} and \ref{sec:sublbJ} that for $\varepsilon$ small enough \[ \N_0\left[\left[\varphi_d(\varepsilon) \val{\range^\varepsilon\cap A_\varepsilon} - \cstap \int_0^\sigma ds \;\ind_A(\hat W_s) \right]^2\right] \leq \csteiii h_d(\varepsilon)^{\theta/2}+2\mcste h_d(\varepsilon)^{\theta}. \] Take $\kappa<\theta/2$ and $\varepsilon_0$ small to get the second upper bound of proposition \ref{th:erange}. \findemo \section{Capacity equivalence for the support and the range of $X$}% super-Brownian motion} \label{sec:cap} \initcste Let $f:(0,\infty )\rightarrow [0,\infty )$ be a decreasing function. We put $f(0)=\lim_{r\downarrow 0} f(r) \in [0,\infty ]$. We define the energy of a Radon measure $\nu$ on $\R^d$ with respect to the kernel $f$ by: $ \ci_f(\nu)=\iint f(\val{x-y}) \nu(dx)\nu(dy)$, and the capacity of a set $\Lambda\in \cb(\R^d)$ by \[ \capaf{\Lambda} =\left[\inf_{\nu(\Lambda)=1} \ci_f(\nu)\right] ^{-1}. \] Following \cite{pp:gwt}, we say that two sets $\Lambda_1$ and $\Lambda_2$ are capacity-equivalent if there exist two positive constants $c$ and $C$ such that for every kernel $f$, we have \[ c\capaf{\Lambda_1} \leq \capaf{\Lambda_2}\leq C\capaf{\Lambda_1} . \] The next lemma is an immediate consequence of the remarks in \cite{pps:tsbm} p.385. \begin{lem} \label{rem:majocapf} Let $\Lambda\subset \R^d$ be a bounded Borel set. Suppose there exist two positive constants $c'$ and $\gamma$ such that \begin{equation*} \lim_{\varepsilon\rightarrow 0} \varepsilon^{\gamma-d}\val{\Lambda^\varepsilon}=c'. \end{equation*} Then there exists a constant $C$ such that for every kernel $f$, we have \[ \capaf{\Lambda} \leq C\left[\int_0^1 f(r) r^{\gamma-1} dr\right]^{-1}. \] \end{lem} \noindent For every measure $\mu\in M_f$, we set \[ S_\varepsilon(\mu)=\iint \mu(dx)\mu(dy)\; p(\varepsilon^2,x-y), \] where $p$ is the Brownian transition density in $\R^d$: $p(t,x)=(2 \pi t)^{-d/2} \expp{-\val{x}^2/2t}$, $(t,x)\in (0,\infty )\times \R^d$. The next lemma is also an immediate consequence of \cite{pps:tsbm} (p.387). \begin{lem} \label{rem:minocapf} Let $\Lambda\subset \R^d$ be a bounded Borel set. Suppose there exist two positive constants $c'$ and $\gamma$ and a measure $\mu\in M_f$ such that $\mu(\Lambda^c)=0$ and \[ \lim_{\varepsilon\rightarrow 0} \varepsilon^{d-\gamma} S_\varepsilon(\mu)=c'. \] Then there exists a constant $c$ such that for every kernel $f$, we have \[ c\left[\int_0^1 f(r) r^{\gamma-1} dr\right]^{-1}\leq \capaf \Lambda . \] \end{lem} For example, for every integer $p\leq d$, we can consider the cube $[0,1]^p$ as a subset of $\R^d$, and then we obviously have \[ \lim_{\varepsilon\rightarrow 0} \varepsilon^{p-d}\val{\left([0,1]^p\right)^\varepsilon}=2 \pi^{(d-p)/2}/\Gamma((d-p)/2), \] and if $\mu$ is Lebesgue measure on $[0,1]^p$, \[ \lim_{\varepsilon\rightarrow 0}\varepsilon^{d-p}S_\varepsilon(\mu)=(2\pi)^{(p-d)/2}. \] Thus we deduce from lemma \ref{rem:majocapf} and \ref{rem:minocapf} that there exist two positive constants $c'_p$, $C'_p$, such that for every kernel $f$, \begin{equation} \label{eq:capr} c'_p \left[\int_0^1 f(r) r^{p-1} dr\right]^{-1}\leq \capaf{[0,1]^p} \leq C'_p \left[\int_0^1 f(r) r^{p-1} dr\right]^{-1}. \end{equation} We shall prove the following result on super-Brownian motion and ISE. \begin{prop} \label{prop:capaequivX} \begin{itemize} \item[(i)] Assume $d\geq 3$. Let $t>0$, $\nu\in M_f$. $\P^X_\nu$-a.s. on $\left\{X_t\neq 0\right\}$, the set $\supp X_t$ is capacity-equivalent to $[0,1]^2$. \item[(ii)] Assume $d\geq 5$. Let $t>0$, $\nu\in M_f$. $\P^X_\nu$-a.s. on $\left\{X_t\neq 0\right\}$, the set $\range_t(X)$ is capacity-equivalent to $[0,1]^4$. Furthermore, if there exists a positive number $\rho<4$ such that $\lim_{\varepsilon\rightarrow 0} \varepsilon^{\rho-d} \val{(\supp \nu)^\varepsilon}=0$, then $\P^X_\nu$-a.s. the set $\range_0(X)$ is capacity-equivalent to $[0,1]^4$. \item[(iii)] Assume $d\geq 5$. The set $\range_t(W)$ is capacity-equivalent to $[0,1]^4$ $\N^{(1)}_0$-a.s. \end{itemize} \end{prop} \noindent \textbf{Proof} of proposition \ref{prop:capaequivX} (i). Let $d\geq 3$. It is well-known that for $t>0$, $\P^X_\nu$-a.s. the set $\supp X_t$ is bounded. Thus, thanks to theorem \ref{th:tribe}, $\P^X_\nu$-a.s., we have \[ \lim_{\varepsilon\rightarrow 0} \varepsilon^{2-d}\val{\left(\supp X_t\right)^\varepsilon} =\cstT (X_t,\ind). \] Now apply lemma \ref{rem:majocapf} to $\Lambda=\supp X_t$, with $\gamma=2$ and take $p=2$ in \reff{eq:capr}. We get that $\P^X_\nu$-a.s., on $\left\{X_t\neq 0\right\}$, there exists a (random) constant $C_1>0$, such that for every kernel $f$, \[ \capaf{\supp X_t}\leq C_1 \capaf{[0,1]^2}. \] For the second part of (i), we use lemma \ref{lem:cvSYt} below. Recall notation $Y_t$ from section \ref{sec:snake}. \begin{lem} \label{lem:cvSYt} Fix $t>0$ and $x\in \R^d$, $d\geq 3$. Then we have \[ \lim_{\varepsilon\rightarrow 0} \varepsilon^{d-2} (2\pi)^{d/2}S_\varepsilon (Y_t)=\frac{4}{d-2} (Y_t,\ind), \] where the convergence holds $\N_x$-a.e. and in $L^2(\N_x)$. \end{lem} \noindent Let us explain how the proof is completed using lemma \ref{lem:cvSYt}. Thanks to lemma \ref{rem:minocapf}, the above lemma and (\ref{eq:capr}) imply that $\N_x$-a.e. on $\left\{Y_t\neq 0\right\}$, there exists a positive constant $c_1$ such that for every kernel $f$, \[ \capaf{\supp Y_t} \geq c_1 \capaf{[0,1]^2}. \] Now remember that for $t>0$, under $\P^X_\nu$, we can write $X_t=\sum_{i\in I}Y_t(W^i)$, where $\sum_{i\in I} \delta_{W^i}$ is a Poisson measure on $C(\R^+,\cw)$ with intensity $\int \nu(dx)\; \N_x[\cdot]$. %Furthermore there is only a finite %number of non zero term in the sum $\sum_{i\in I}Y_t(W^i)$. On $\left\{X_t\neq 0\right\}$, there exists $i_0$ such that $Y_t(W^{i_0})\neq 0$. Then we have $\supp Y_t(W^{i_0}) \subset \supp X_t$. Thus the previous lemma entails that there exists a.s. a positive constant $c_1(W^{i_0})$ such that for every kernel $f$, \[ \capaf{\supp X_t}\geq \capaf{\supp Y_t(W^{i_0})}\geq c_1(W^{i_0}) \capaf{[0,1]^2} \] This completes the proof of (i). \findemo \noindent \textbf{Proof} of proposition \ref{prop:capaequivX} (ii). Let $d\geq 5$. We argue as in the proof of (i) using theorem \ref{th:rangeX} instead of theorem \ref{th:tribe} and the following lemma instead of lemma \ref{lem:cvSYt}. \begin{lem} \label{lem:cvSrYt} Fix $t\geq 0$ and $x\in \R^d$, $d\geq 5$. Then we have for every $T>t\geq 0$, \[ \lim_{\varepsilon\rightarrow 0} \varepsilon^{d-4} (2\pi)^{d/2} S_\varepsilon \left(\int_t^T ds\;Y_s \right) =\frac{16}{(d-2)(d-4)} \int_t^T ds \;(Y_s ,\ind), \] where the convergence holds $\N_x$-a.e. and in $L^2(\N_x)$. \end{lem} \findemo \noindent \textbf{Proof} of proposition \ref{prop:capaequivX} (iii). Let $d\geq 5$. For the first part we argue as in the proof of (i) using the second part of corollary \ref{cor:cvpsrangeY} instead of theorem \ref{th:tribe}. Notice that thanks to \reff{eq:Nnormal} and the scaling property of the family $(\N^{(r)}_0,r>0)$, the convergence in lemma \ref{lem:cvSrYt} also holds $\N^{(1)}_0$-a.s. The second part of (iii) is then a direct consequence of lemma \ref{rem:minocapf} (with $\mu=\int_0^T ds\; Y_s$ and $\gamma=4$) and \reff{eq:capr} (with $p=4$). \findemo \noindent The proofs of lemma \ref{lem:cvSYt} and lemma \ref{lem:cvSrYt} are very similar. We shall only prove the latter. The former uses the same techniques in a simpler way. \noindent \textbf{Proof} of lemma \ref{lem:cvSrYt}. We first want to show the convergence in $L^2(\N_x)$. Fix $T>t\geq 0$. By standard monotone class arguments, we deduce from the results of section \ref{sec:annexemo} an explicit expression for \[ \N_x\left[\int_0^T \cdots \int_0^T ds_1\ldots ds_4\; \int\cdots \int Y_{s_1}(dx_1)\ldots Y_{s_4}(dx_4) g(s_1,\ldots,s_4,x_1,\ldots,x_4)\right], \] where $g$ is any measurable positive function on $(\R^+)^4\times (\R^d)^4$. Specializing to the case $g(s_1,\ldots,s_4,x_1,\ldots,x_4)=\prod_{i=1}^4 \ind_{[t,T]}(s_i) p(\varepsilon^2, x_1-x_2) p(\varepsilon^2, x_3-x_4)$, we get \begin{multline*} \N_x \left[S_\varepsilon\left(\int_t^T ds\;Y_s\right)^2\right]\\ \begin{aligned}[b] &= \inv{3} 4!2^3 \int_0^T ds \int dy \;p(s,x-y) \Bigg\{ 4\int^{T-s}_{(t-s)_+} ds_1\int dy_1\; p(s_1,y-y_1)\int_0^{T-s}ds_2\\ &\hspace{1.5cm}\int dy_2\; p(s_2,y-y_2)\int_{(t-s-s_2)_+}^{T-s-s_2} ds_3\int dy_3 \; p(s_3,y_2-y_3)\int_0^{T-s-s_2} ds_4\\ &\hspace{1.5cm}\int dy_4 \; p(s_4,y_2-y_4) %\\ %&\hspace{1.5cm} \int_{(t-s-s_2-s_4)_+}^{T-s-s_2-s_4} ds_5\int dy_5 \; p(s_5,y_4-y_5)\\ &\hspace{1.5cm}\int_{(t-s-s_2-s_4)_+}^{T-s-s_2-s_4} ds_6\int dy_6 \; p(s_6,y_4-y_6) \\ &\hspace{3cm}[p(\varepsilon^2,y_1-y_3)p(\varepsilon^2,y_5-y_6)+ p(\varepsilon^2,y_1-y_5)p(\varepsilon^2,y_3-y_6)\\ &\hspace{7cm}+p(\varepsilon^2,y_1-y_6)p(\varepsilon^2,y_3-y_5)] \\ &\hspace{1cm}+ \int_0^{T-s}ds_7\int dy_7\; p(s_7,y-y_7)\int_{(t-s-s_7)_+}^{T-s-s_7}ds_8\int dy_8\; p(s_8,y_7-y_8)\\ &\hspace{1.cm}\phantom{+ \int_0^{T-s}ds_7\int dy_7\; p(s_7,y-y_7)} \int_{(t-s-s_7)_+}^{T-s-s_7} ds_9\int dy_9 \; p(s_9,y_7-y_9)\\ &\hspace{1.5cm}\int_0^{T-s } ds_{10}\int dy_{10} \; p(s_{10},y-y_{10}) \int_{(t-s-s_{10})_+}^{T-s-s_{10}} ds_{11}\int dy_{11} \; p(s_{11},y_{10}-y_{11})\\ &\hspace{1.5cm} \phantom{\int_0^{t-s } ds_{10}\int dy_{10} \; p(s_{10},y-y_{10})} \int_{(t-s-s_{10})_+}^{T-s-s_{10}} ds_{12}\int dy_{12} \; p(s_{12},y_{10}-y_{12}) \\ \end{aligned}\\ [ p(\varepsilon^2,y_{8}-y_{9})p(\varepsilon^2,y_{11}-y_{12})+ p(\varepsilon^2,y_{8}-y_{11})p(\varepsilon^2,y_{ 9}-y_{12})\\ + p(\varepsilon^2,y_{8}-y_{12})p(\varepsilon^2,y_{ 9}-y_{11})] \Bigg\} \end{multline*} We write $J_1$, $J_2$, $J_3$, $J_4$, $J_5$, and $J_6$, respectively for the integrals corresponding to the integrands $p(\varepsilon^2,y_1-y_3)p(\varepsilon^2,y_5-y_6)$, $p(\varepsilon^2,y_1-y_5)p(\varepsilon^2,y_3-y_6)$, $p(\varepsilon^2,y_1-y_6)p(\varepsilon^2,y_3-y_5)$, $p(\varepsilon^2,y_{8}-y_{9})p(\varepsilon^2,y_{11}-y_{12})$, $p(\varepsilon^2,y_{8}-y_{11})p(\varepsilon^2,y_{ 9}-y_{12})$, and $p(\varepsilon^2,y_{8}-y_{12})p(\varepsilon^2,y_{ 9}-y_{11})$ respectively. As we shall see the integral $J_4$ gives the main contribution. Before proceeding to the calculations, we give three useful bounds: for every positive real numbers $s$, $\varepsilon^2<2^{-1}(T^{-1}\wedge T)$, we have for $d\geq 5$ \begin{align} \label{eq:2-d} \int_0^T \left(\varepsilon^2+s+r\right)^{-d/2}dr &\leq \frac{2}{d-2}\left(\varepsilon^2+s\right)^{1-d/2}, \\ \label{eq:4-d} \int_0^T \left(\varepsilon^2+s+r\right)^{1-d/2}dr &\leq \frac{2}{d-4}\left(\varepsilon^2+s\right)^{2-d/2}, \\ \label{eq:6-d} \int_0^T \left(\varepsilon^2+r\right)^{2-d/2}dr &\leq H_T(\varepsilon):= \begin{cases} {2}{(d-6)^{-1}} \varepsilon^{6-d} & \text{if $d\geq 7$},\\ 4\ln {\varepsilon}^{-1} & \text{if $d=6$},\\ \displaystyle \sqrt{6T} & \text{if $d=5$}. \end{cases} \end{align} From now on, we assume that $\varepsilon^2<2^{-1}(T^{-1}\wedge T)$ and also $\varepsilon^2\ln \varepsilon^{-1} 0$ and $J$ a nonnegative measurable function on $\R^+\times \R^d$, such that $M_T=\sup_{[0,T]\times \R^d} J(t,x)<\infty $. We define the family of measurable functions $(h_n,n\geq 1)$ on $\R^+\times \R^d$, by the initial condition \begin{align} \nonumber h_1(t)&=J(t),\\ \intertext{and the recurrence} \label{eq:rechn} h_n(t)&=2\sum_{k=1}^{n-1}\int_0^t ds\; P_s\left[h_k(t-s) h_{n-k}(t-s)\right]\quad \text{for} \quad n\geq 2 . \end{align} We clearly have for every $n\geq 1$, \[ \sup_{[0,T]\times \R^d} \val{h_n}\leq [4T]^{n-1} [2M_T]^n \gamma_n. \] Thus the power series $w(\lambda,t)=\sum (-1)^{n+1}\lambda^n h_n(t)$ is normally convergent on $[0,T]\times \R^d$ for $\val{\lambda}<[8TM_T]^{-1}$. And it clearly solves the integral equation on $[0,T]\times \R^d$ \begin{equation} \label{eq:vlambda} w(t)+2\int_0^t ds\; P_s \left[w(t-s)^2\right]=\lambda J(t). \end{equation} To get the uniqueness of the solution to the previous integral equation, use arguments similar to Gronwall's lemma. Finally we can compute the moments for the process $Y$ under $\N_x$. Indeed, let $\varphi\in \cb_{b+} (\R^+\times \R^d)$. We have shown that for $\lambda>0$, the function $v_\lambda(t,x)=\N_x \left[1-\exp{-\lambda \int_0^t (Y_{t-s},\varphi(s))} ds\right]$ is the unique solution to (\ref{eq:vlambda}) on $\R^+\times \R^d$ with $J(t,x)=\int_0^t ds\; P_{s} [\varphi(t-s)](x)$. Thus for $\lambda\geq 0$ small enough, we have $v_\lambda(t)=w(\lambda,t)$. Then from the series expansion for $w(\lambda,t)$, we get for every integer $n\geq 1$ \[ \N_x\left[\left(\int_0^t ds\; (Y_{t-s},\varphi(s))\right)^n\right]= n!h_n(t,x), \] where the functions $h_n$ are defined by $h_1(t)=\int_0^t ds\; P_{s}[\varphi(t-s)]$, and the recurrence (\ref{eq:rechn}). In the same way it can be shown that for every $\varphi\in \cb_{b+}(\R^d)$, for every $t\geq 0$, $n\geq 1$, \[ \N_x\left[(Y_{t},\varphi)^n\right]= n!h_n(t,x), \] where the functions are defined by $h_1(t)=P_{t}[\varphi]$, and the recurrence (\ref{eq:rechn}). \subsection{Some properties of the function $u_1$} \label{sec:annexeu} We consider the function $u_1$, which is the maximal solution on $(1,\infty )$ of the non linear differential equation \[ {u}''(r)+\frac{d-1}{r} u' (r)=4u(r)^2. \] \begin{lem} \label{lem:a0} There exist positive constants $\csta$, $\cstb$ and $\cstaaa$, depending only on $d$, such that \begin{align} \nonumber \lim_{r \rightarrow \infty }r^{d-2}u_1(r)=\csta\quad \text{if $d\geq 5$}, \quad & \lim_{r \rightarrow \infty }r^2\log(r)\;u_1(r) =\csta=1/2 \quad \text{if $d=4$};\\ \intertext{furthermore for every $r>1$,} \label{eq:umino1} u_1(r) \geq \csta r^{2-d}\quad \text{if $d\geq 5$}, \quad & u_1(r) \geq \csta r^{-2} \log (2r)^{-1}\quad \text{if $d=4$}; \intertext{and for every $r\geq 4/3$,} \label{eq:umajo1} u_1(r)\leq \cstb r^{2-d}\quad \text{if $d\geq 5$}, \quad & u_1(r)\leq \cstb [2 r^2 \log (r)]^{-1}\quad \text{if $d=4$}; \end{align} \begin{align} \label{eq:udel1} u_1(r)&\leq \csta r^{2-d}+\cstaaa r^{6-2d}\quad\text{if $d\geq 5$},\\ \label{eq:udel41} u_1(r)&\leq \csta r^{-2} \log (r)^{-1}+\cstaaa r^{-2} \log (r)^{-2}\log(\log (r))\quad \text{if $d=4$}. \end{align} \end{lem} We will prove this lemma by giving the asymptotic expansion of $u_1$ at $\infty $. For $d\geq 5$, we will see the constant $\csta$ can be expressed as the radius of convergence of a series. We introduce the auxiliary function \[ z(t)=4(d-2)^{-2(d-1)/(d-2)} t\; u_1\left[\left(\frac{t}{d-2}\right)^{1/(d-2)}\right], \quad \text{for } t>d-2. \] This function is solution on $(d-2,\infty )$ of \begin{equation} \label{eq:diffy} y''(t) =t^{-\delta-2} y(t)^2, \end{equation} where $\displaystyle \delta=\frac{d-4}{d-2}$. We deduce from \cite{t:abs} p.132 that the function $z(t)$ (i.e. $r^{d-2}u_1(r)$) is decreasing. \noindent \textbf{Proof} in the case $d\geq 5$. We deduce from theorem 1.1 of \cite{t:abs} that the limit $q=\lim_{t\rightarrow \infty } z(t)$ exists and is positive. Hence by integrating (\ref{eq:diffy}) twice from $t$ to $\infty $, we get for $t>d-2$, \begin{equation} \label{eq:integz} z(t) -q =\int_t^\infty (r-t) r^{-\delta-2} z(r)^2 dr. \end{equation} Now consider the sequence $(q_n,n\geq 0)$ defined by $q_0=1$ and the recurrence \[ q_{n}=\inv{n\delta (n\delta+1)}\sum_{k=0}^{n-1} q_kq_{n-k-1} , \quad \text{for}\quad n\geq 1. \] Clearly we have for every $n\geq 0$, $q_n\leq 2\left[4/\delta\right]^{n} \gamma_{n+1},$ where the sequence $(\gamma_n,n\geq 1)$ has been introduced in section \ref{sec:annexemo}. The power series $\sum q_n q^{n+1} t^{-\delta n}$ is convergent and even $C^\infty $ as a function of $t$ at least for $t>t_1=\left[4q/\delta\right]^{1/\delta}$. This power series also solves (\ref{eq:integz}) for $t>t_1$. The same arguments as in the proof of the Gronwall lemma show that equation (\ref{eq:integz}) possesses a unique solution bounded in a neighborhood of infinity. Thus the function $z$ and the power series agree for $t>t_1$. Since the function $z$ is analytic on $(d-2,\infty )$ and since $q$ and the coefficients $q_n$ are positive, we deduce that the radius of convergence of the series $\sum q_n s^n$ is $q (d-2)^{-\delta}$ and that for $t>d-2$ \[ z(t) =\sum_{n=0}^\infty q_n q^{n+1} t^{-n\delta}. \] Thus we get with obvious notation for $r>1$, \begin{align*} u_1(r) &= 4^{-1} (d-2)^{d/(d-2)} r^{2-d}\sum_{n=0}^\infty q_n q^{n+1} (d-2)^{-n(d-4)/(d-2)} r^{-n(d-4)}\\ &= r^{2-d}\sum_{n=0}^\infty \delu_n r^{-n(d-4)}. \end{align*} Since the function $r^{d-2} u_1(r)$ is decreasing we get \reff{eq:umino1}. Since the real numbers $(\delu_n, n\geq 0)$ are positive, \reff{eq:umajo1} and \reff{eq:udel1} follow easily. Notice that $4 (d-2)^{-2} \csta$ is the radius of convergence of the series $\sum q_n s^n$. \findemo \noindent \textbf{Proof} in the case \textbf{$d=4$}. We write $f(t)\thicksim g(t)$ at $0+$ when the real function $f$ and $g$ are positive or negative on $I=(0,0+\varepsilon)$ for some $\varepsilon>0$ and $\lim_{t\in I, t\rightarrow 0} f(t)/g(t)=1$. We also write $f(t)\thicksim g(t)$ at $\infty $ when $f(1/t)\thicksim g(1/t)$ at $0+$. We know from \cite{t:abs} p.133 that $z(t)\thicksim \log (t)^{-1}$ at $\infty $. We deduce from \reff{eq:diffy} that $z'(t)$ is negative on $(2,\infty )$ and $z''(t)\thicksim [t\log (t)]^{-2}$ at $\infty $. By integration, we get $z'(t)\thicksim t^{-1}\log (t)^{-2}$ at $\infty $. We now consider the function $w(s)=z(\expp{s})$ which solves $w''-w'=w^2$ on $(\log 2,\infty )$. Notice that the function $w$ is positive decreasing and $w'$ is negative. We also have $w(s)\thicksim s^{-1}$, $w'(s)\thicksim -s^{-2}$ and $w''(s)=o(s^{-2})$ at $\infty$. Thus the function defined on $(0,\infty )$ by \[ p(w(s))=w'(s), \quad \text{for}\quad s\in (\log 2, \infty ), \] is well defined and even of class $C^1$, and $p'(w(s))=w''(s)/w'(s)$. Thus the function $p$ can be extended as a $C^1$ function on $[0,\infty )$ by setting $p(0)=0$ and $p'(0)=0$. Furthermore it solves \[ p(w)p'(w)-p(w)=w^2 \quad \text{on}\quad [0,\infty ). \] We also have $p(w)\thicksim -w^2$ at $0+$. We consider the sequence $(\rho_n,n\geq 2)$ defined by $\rho_2=1$ and the recurrence \[ \rho_n=\sum_{k=2}^{n-1} k \rho_k\rho_{n-k+1},\quad \text{for} \quad n\geq 3. \] The radius of convergence of the series $\sum (-1)^{n+1} \rho_n w^n$ is $0$, nevertheless we will prove it is the asymptotic expansion of $p$ at $0+$. We set $H_n(w)=\sum_{k=2}^n (-1)^{k+1} \rho_k w^k$ for $n\geq 2$. We now prove by recurrence that $p(w)=H_n(w)+h_n(w)$, where $h_n(w)=o(w^n)$ at $0+$. This is true for $n=2$. Let us assume it is true at stage $n$. Let $g_{n,\alpha}(w)= (1-\alpha)(-1)^{n}\rho_{n+1} w^{n+1} -h_n(w)$. We easily have \begin{align*} g'_{n,\alpha}(w)p(w)+g_{n,\alpha}(w)[H'_n(w)-1] &=\alpha(-1)^n \rho_{n+1}w^{n+1}+o(w^{n+1}),\\ &=(-1)^{n+1} \rho_{n+2}w^{n+2}+o(w^{n+2}), \quad \text{if} \quad \alpha=0. \end{align*} Let us assume $n$ is even. For $\alpha=0$, the above right hand side is negative on $(0,\varepsilon]$, for $\varepsilon$ small enough. Since $p$ is negative and $[H'_n(w)-1]<0$ on $[0,\varepsilon]$, for $\varepsilon$ small, we see that $g_{n,0}(w)<0$ implies $g_{n,0}'(w)\geq 0$. As $g_{n,0}(0)=0$, we get by contradiction that $g_{n,0}\geq 0$ on $[0,\varepsilon]$. This implies $h_n(w)\leq \rho_{n+1} w^{n+1}$. Similar arguments for $\alpha>0$ implies that $g_{n,\alpha}\leq 0$ on $[0,\varepsilon_\alpha]$ for $\varepsilon_\alpha>0$ small enough. Since this holds for any $\alpha>0$ and since $h_n(w)\leq \rho_{n+1} w^{n+1}$ for $w$ small enough, we deduce that $h_{n+1}(w)=h_n(w)-\rho_{n+1} w^{n+1}=o(w^{n+1})$. If $n$ is odd the proof is similar. From the definition of $p$, we then have $w'(s)=H_n(w(s))+O(w(s)^{n+1})$ at $\infty $. For $n=3$ this gives $w'(s)=-w(s)^2+2 w(s)^3+O(w(s)^4)$ at $\infty $. Since $w(s)\thicksim s^{-1}$ at $+\infty $, we deduce by integration that \[ \inv{w(s)}-2 \log w(s) + O(1)=s \text{ at infinity}. \] Standard arguments yields $w(s)=s^{-1}+2 s^{-2} \log (s) + O(s^{-2})$ at infinity. Thus we have \[ u_1(r)=\inv{r^2} \left[\inv{2\log (r)} +\frac{\log (\log (r))}{4\log (r)^2} +O\left(\log (r)^{-2} \right)\right] \text{ at } +\infty . \] Notice the previous calculation can be continued to give an asymptotic expansion of $u_1$ at infinity. The inequalities \reff{eq:umajo1} and \reff{eq:udel41} follow easily. We will now prove that for every $r>1$, $u_1(r) \geq [2 r^2 \log (2r)]^{-1}$. We consider the function $w(r)=u_1(r)-[2r^2 \log(2r)]^{-1}$. The function $w$ is positive at least over $(1,1+\eta)\cap(\eta^{-1}, \infty )$ for $\eta$ small. Let us assume that $w$ achieves its minimum at $r_0$ and that $w(r_0)\leq 0$. Then we have $r_0\in [1+\eta,\eta^{-1}]$, $w'(r_0)=0$ and $w''(r_0)\geq 0$. An easy computation gives \[ w''(r)=4w(r)\left[u_1(r)+\inv{2r^2\log (2r)}\right]-\frac{3}{r} w'(r)-\inv{2 r^4 (\log (2r))^3}. \] Evaluation at $r=r_0$ implies that $w''(r_0)<0$. This contradicts the assumption. Hence $w$ is positive, that is we get \reff{eq:umino1} for $d=4$. \findemo \begin{thebibliography}{10} \bibitem{a:tbm} D.~ALDOUS. \newblock Tree based models for random distribution of mass. \newblock {\em J. Statist. Phys.}, 73(3-4):625--641, 1993. \bibitem{b:emp} R.~BLUMENTHAL. \newblock {\em Excursions of Markov processes}. \newblock Birkh\"auser, Boston, 1992. \bibitem{ds:smb} E.~DERBEZ and G.~SLADE. \newblock The scaling limit of lattice trees in high dimension. \newblock Preprint, 1996. \bibitem{ds:ltsbm} E.~DERBEZ and G.~SLADE. \newblock Lattice trees and super-{B}rownian motion. \newblock {\em Canad. Math. Bull.}, 40(1):19--38, 1997. \newblock To appear. \bibitem{d:rfsmsi} E.~DYNKIN. \newblock Representation for functionals of superprocesses by multiple stochastic integrals, with application to self-intersection local times. \newblock {\em Ast\'erisque}, 157-158:1147--1171, 1988. \bibitem{d:bpss} E.~DYNKIN. \newblock Branching particle systems and superprocesses. \newblock {\em Ann. Probab.}, 19:1157--1194, 1991. \bibitem{d:pacnde} E.~DYNKIN. \newblock A probabilistic approach to one class of nonlinear differential equations. \newblock {\em Probab. Th. Rel. Fields}, 89:89--115, 1991. \bibitem{d:ibmp} E.~DYNKIN. \newblock {\em An introduction to branching measure-valued processes}, volume~6 of {\em CRM Monograph series}. \newblock Amer. Math. Soc., Providence, 1994. \bibitem{lg:pvmp} J.-F. LE~GALL. \newblock A class of path-valued {M}arkov processes and its applications to superprocesses. \newblock {\em Probab. Th. Rel. Fields}, 95:25--46, 1993. \bibitem{lg:urtbe} J.-F. LE~GALL. \newblock The uniform random tree in a {B}rownian excursion. \newblock {\em Probab. Th. Rel. Fields}, 96:369--383, 1993. \bibitem{lg:hppt} J.-F. LE~GALL. \newblock Hitting probabilities and potential theory for the {B}rownian path-valued process. \newblock {\em Ann. Inst. Four.}, 44:277--306, 1994. \bibitem{lg:pvmppde} J.-F. LE~GALL. \newblock A path-valued {M}arkov process and its connections with partial differential equations. \newblock In {\em Proceedings in First European Congress of Mathematics}, volume~II, pages 185--212. Birkh\"auser, Boston, 1994. \bibitem{lg:pccm} J.-F. {\sortnoop{LE GALLL}}LE~GALL. \newblock Communication personnelle. \bibitem{pp:gwt} R.~PEMANTLE and Y.~PERES. \newblock Galton-{W}atson trees with the same mean have the same polar sets. \newblock {\em Ann. Probab.}, 23(3):1102--1124, 1995. \bibitem{pps:tsbm} R.~PEMANTLE, Y.~PERES, and J.~W. SHAPIRO. \newblock The trace of spatial {B}rownian motion is capacity-equivalent to the unit square. \newblock {\em Probab. Th. Rel. Fields}, 106(3):379--400, 1996. \bibitem{p:smpssbm} E.~A. PERKINS. \newblock The strong {M}arkov property of the support of super-{B}rownian motion. \newblock In {\em The {D}ynkin Festschrift}, volume~34 of {\em Progr. Probab.}, pages 307--326, Boston, 1994. Birkh\"auser. \bibitem{ps:bm} S.~C. PORT and C.~J. STONE. \newblock {\em Brownian motion and classical potential theory}. \newblock Academic Press, 1978. \bibitem{t:abs} S.~D. TALIAFERRO. \newblock Asymptotic behavior of solutions of $y''=\phi (t)\; y^\lambda$. \newblock {\em J. Math. Analys. and Appl.}, 66:95--134, 1978. \bibitem{t:rsbm} R.~TRIBE. \newblock A representation for super-{B}rownian motion. \newblock {\em Stoch. Process. and Appl.}, 51:207--219, 1994. \end{thebibliography} \end{document}