\documentclass[11pt]{amsart} \usepackage{amsmath,amssymb,amsthm} \textwidth=15cm \setlength{ \hoffset}{-1cm} \newcommand{\sortnoop}[1]{} \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}}}} \newcommand{\Capa}{\mathop{\mathrm{Cap}}\nolimits} \newcommand{\capaD}{\mathop{\mathrm{cap_D}}\nolimits} \newcommand{\normt}[1]{\mathop{\parallel\! #1 \! \parallel}\nolimits} \begin{document} \title[Characterization of G-regularity for super-Brownian motion]{Characterization of G-regularity for super-Brownian motion and consequences for parabolic partial differential equations} \date{\today} \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 of the first author was done at the \'Ecole Nationale des Ponts et Chauss\'ees and at MSRI, supported by NSF grant DMS-9701755.} \author{Jean-St\'ephane Dhersin} \address{UFR de Math\'ematiques et d'Informatique, Universit\'e Ren\'e Descartes, Paris, France} \email{dhersin@math-info.univ-paris5.fr} \begin{abstract} We give a characterization of G-regularity for super-Brownian motion and the Brownian snake. More precisely, we define a capacity on $E=(0,\infty )\times \R^d$, which is not invariant by translation. We then prove that the hitting probability of a Borel set $A\subset E$ for the graph of the Brownian snake starting at $(0,0)$ is comparable, up to multiplicative constants, to its capacity. This implies that super-Brownian motion started at time $0$ at the Dirac mass $\delta_0$ hits immediately $A$ (that is $(0,0)$ is G-regular for $A^c$) if and only if its capacity is infinite. As a direct consequence, if $Q\subset E$ is a domain such that $(0,0)\in \partial Q$, we give a necessary and sufficient condition for the existence on $Q$ of a positive solution of $\partial_t u+\frac{1}{2}\Delta u =2u^2$ which blows up at $(0,0)$. We also give an estimation of the hitting probabilities for the support of super-Brownian motion at fixed time. We prove that if $d\geq 2$, the support of super-Brownian motion is intersection-equivalent to the range of Brownian motion. \end{abstract} %\vspace{0.7cm} %\noindent\textit{Mathematics Subject Classification} 1991: 60G57, 35K60. \keywords{Super-Brownian motion, Brownian snake, G-regularity, para\-bolic nonlinear PDE, hitting probabilities, capacity, intersection-equivalence.} \subjclass{60G57, 35K60.} \maketitle \section{Introduction} The purpose of this paper is to give a characterization of the so called G-regularity for super-Brownian motion introduced by Dynkin~\cite{d:spde}. Let us recall that a point $(r,x)\in {\mathbb R}\times{\mathbb R}^d$ is G-regular for an open set $Q\subset {\mathbb R}\times{\mathbb R}^d$ if a.s. the graph of a super-Brownian motion started at time $r$ with the Dirac mass at $x$ immediately intersects $Q^c$, the complementary of $Q$. (This definition can be extended to any Borel set.) We also recall that this is equivalent to the existence of nonnegative solutions of the equation $\frac{\partial u}{\partial t}+\frac{1}{2}\Delta u=2\,u^2$ on the open set $Q$, which blow up at $(r,x)\in \partial Q$. Let $E=(0,\infty )\times{\mathbb R}^d$. We prove that $(0,0)$ is $G$-regular for a Borel set $A\subset \R\times \R^d$ if and only if the capacity of $A^c\cap E$ is infinite, for the following capacity: for any Borel set $A'\subset E$, \[ \capa (A')=\left[\inf I(\nu)\right]^{-1}, \quad \text{where } \] \[ I(\nu)= \iint_E dsdy\; p(s,y)\left(\iint_E \nu(dt,dx)\frac{p(t-s,x-y) }{p(t,x)}\right)^2, \] and $p$ denotes the heat kernel: \[ p(t,x)=\left\{ \begin{array}{ll} (2 \pi t)^{-d/2}\expp{-\val{x}^2/2t} \quad &\text{ if } (t,x)\in E,\\ 0 &\text{ if } (t,x)\in (-\infty ,0]\times \R^d. \end{array} \right. \] ($\val{\cdot}$ denotes the Euclidean norm on $\R^d$.) The infimum is taken over all probability measures $\nu$ on $E$ such that $\nu(A')=1$. Notice this capacity is not invariant by translation in time or space. This capacity arises naturally when one consider the Brownian snake, a useful tool to study super-Brownian motion. Indeed, using potential theory of symmetric Markov process, $I(\nu)$ can be viewed as the energy, with respect to the Brownian snake, of a certain probability measure (see section 4 for more details). We extend a result due to Dhersin and Le~Gall~\cite{dlg:ktsbm} where the authors study $G$-regularity of $(0,0)$ for sets $Q=\{(s,y)\in E;\,|y|<\sqrt{s}\,h(s)\}$, where $h$ is a positive decreasing function defined on $(0,\infty )$. Our result can also be viewed as a parabolic extension of the Wiener's test proved by Dhersin and Le~Gall~\cite{dlg:wtsbmbs} in an elliptic setting. The proof of our results relies on the Brownian snake introduced by Le~Gall. We only give definition and some properties for completeness in this paper, and refer to Le~Gall~\cite{lg:pvmp,lg:pvmppde} for a detailed presentation. We will use time inhomogeneous notations. Let $(r,x)\in {\mathbb R}\times {\mathbb R}^d$ be a fixed point. We denote by ${\mathcal W}_{r,x}$ the set of all stopped paths in ${\mathbb R}^d$ started at $x$ at time $r$. An element ${\rm w}$ of ${\mathcal W}_{r,x}$ is a continuous mapping ${\rm w}:[r,\zeta]\to{\mathbb R}^d$ such that ${\rm w}(r)=x$, and $\zeta=\zeta_{(\rm w)}\in [r,\infty )$ is called its lifetime. We denote by $\hat \rw $ the end point $\rw(\zeta)$. With the metric $d(\rw,\rw')=\val{\zeta_{(\rw)} -\zeta_{(\rw')}}+\sup_{s\geq r}\val{\rw(s\wedge\zeta_{(\rw)})-\rw'(s\wedge\zeta_{(\rw')})}$, the space $\cw_{r,x}$ is a Polish space. The Brownian snake started at $x$ at time $r$ is a continuous strong Markov process $W=(W_s,s\geq 0)$ with values in $\cw_{r,x}$, 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 $[r,\infty )$. \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, such that for $s'\geq s$: \begin{itemize} \item[$\cdot$] $W_{s'}(t)=W_s(t)$ for $r\leq t\leq m(s,s')={{\displaystyle } \inf_{v\in [s,s']}\zeta_v}$. \item[$\cdot$] $(W_{s'}(m(s,s')+t)-W_{s'}(m(s,s')),0\leq t\leq \zeta_{s'}-m(s,s'))$ is a Brownian motion in ${\mathbb R}^d$ independent of $W_s$. \end{itemize} \end{itemize} From now on we shall consider the canonical realization of the process $W$ defined on the space $\Omega=C(\R^+,\cw_{r,x})$, and denote by $\ce_{\rw}$ the law of $W$ started at $\rw\in\cw_{r,x}$. The trivial path $\textbf{x}_r$ such that $\zeta_{(\textbf{x}_r)}=r$, $\textbf{x}_r (r)=x$ is clearly a regular point for the process $\left(W, \ce_{\rw}\right)$. We denote by $\N_{r,x}$ the excursion measure outside $\left\{\textbf{x}_r\right\}$. Notice that $\N_{r,x}$ is an infinite measure. The distribution of $W$ under $\N_{r,x}$ can be characterized as above, except that now the lifetime process $\zeta$ is distributed according to the It\^o measure of excursions of linear reflecting Brownian motion in $[r,\infty )$. We normalize $\N_{r,x}$ so that, for every $\varepsilon>0$, \[ \N_{r,x}\left[\sup_{s\geq 0}\zeta_s>\varepsilon+r\right]=\inv{2\varepsilon}. \] Let $\sigma=\inf\left\{s>0;\zeta_s=r\right\}$ denote the duration of the excursion of $\zeta$ under $\N_{r,x}$. The graph $\cg^*$ of $W$ is defined under $\N_{r,x}$ by \[ \cg^*=\left\{(t,W_s(t)); r< t\leq \zeta_s, 0< s < \sigma\right\} = \left\{(\zeta_s,\hat W_s); 0 r,s\geq 0\right)$ of the local time of $\zeta$ at level $t$ and time $s$, and define the measure valued process $Y$ on $\R^d$ by setting for every $t>r$, for every $\varphi\in \cb_{b+}(\R^d)$, \[ (Y_t,\varphi)=\int_0^\sigma dl_s^{t} \; \varphi(\hat W_s). \] Let $\cw_{r}=\bigcup _{x\in \R^d}\cw_{r,x}$. Let $\mu$ be a finite measure on $\R^d$, and $\sum_{i\in I} \delta_{W^i}$ be a Poisson measure on $C(\R^+,\cw_{r})$ with intensity $\int\mu(dx)\N_{r,x}[\cdot]$. Then the process $X$ defined by $X_r=\mu$ and $X_t=\sum_{i\in I}Y_t(W^i)$ if $ t>r$, is a super-Brownian motion started at time $r$ at $\mu$ (see \cite{lg:pvmp,lg:pvmppde}). We shall denote by $\PP_{r,\mu}$ (resp. $\PP_{r,x}$) the law of the super-Brownian motion started at time $r$ at $\mu$ (resp. at the Dirac mass $\delta_x$). We deduce from the normalization of $\N_{r,x}$ that, for every $t>r$, $\N_{r,x} \left[Y_t\neq 0\right]=1/2(t-r)<\infty $. This implies that there is only a finite number of indices $i\in I$ such that $\cg^*(W^i)\cap [t,\infty )\times \R^d$ is non empty for $t>r$. We consider the graph of $X$: \label{com:cgX} \[ \cg(X)=\bigcup_{\varepsilon>r}\left(\overline{\bigcup_{t\geq \varepsilon} \{t\}\times \supp X_t }\right)=\bigcup _{i\in I} \cg^*(W^i), \] where $\bar A$ denotes the closure of $A$. A set $A\subset \cb(\R\times \R^d)$ is called $G$-polar if $\PP_{r,x} [\cg(X)\cap A\neq \emptyset]=0$ for every $(r,x)\in \R\times \R^d$. From Poisson measure theory, we have \begin{equation*} %\label{eq:pp} \PP_{r,x} [\cg(X)\cap A\neq \emptyset]=1-\expp{-\N_{r,x}[\cg^*\cap A\neq \emptyset]}. \end{equation*} Hence $A$ is $G$-polar if and only if $\N_{r,x}[\cg^*\cap A\neq \emptyset]=0$ for all $(r,x)\in \R\times \R^d$. We consider the capacity defined by: for $A\in \cb(\R\times \R^d)$, \[ \capa' (A)=\left[\inf \iint_{\R\times \R^d} dsdy \left(\iint_{(s,\infty )\times \R^d} \nu(dt,dx)p(t-s,x-y)\expp{-(t-s)/2 }\right)^2\right]^{-1}, \] where the infimum is taken over all probability measures $\nu$ on $\R\times \R^d$ such that $\nu(A)=1$. Dynkin proved (see Theorem 3.2 in \cite{d:sap}) that $A\in \cb(\R\times \R^d)$ is $G$-polar if and only if $\capa' (A)=0$. (We have $\capa' (A)=0 \Leftrightarrow \N_{r,x}[\cg^*\cap A\neq \emptyset]=0$ for all $(r,x)\in \R\times \R^d$.) It is easy to check that if $A\subset E$ is a compact set then \[ \capa' (A)=0 \Leftrightarrow \capa (A)=0. \] This can be extended to all Borel subsets of $E$ since the two capacities are inner capacities (see Meyers \cite{m:tcpf}). In fact it seems more relevant to consider the capacity $\capa$ to characterize $G$-regularity, as we shall see. We have the following quantitative theorem. \begin{theorem} \label{th:ucap} There exists a constant $C_0$ such that for any $A\in \cb(E)$, \[ 4^{-1} \capa(A)\leq \N_{0,0}[\cg^*\cap A\neq \emptyset]\leq C_0 \capa(A). \] \end{theorem} The proof of Theorem~\ref{th:ucap} is split in two parts. In section 2, we introduce a capacity associated with a weighted Sobolev space, which is equivalent to the capacity $\capa$. In section 3, using the connections between super-Brownian motion and partial differential equations, we prove the upper bound with this new capacity, and hence for the capacity $\capa$. The lower bound is obtain in section 4, by using additive functionals of the Brownian snake introduced in \cite{dlg:wtsbmbs}. Now, for $A\in \cb( \R\times \R^d)$, we consider under $\PP_{r,x}$ the random time \[ \tau_A=\inf\{t>r, \left(\{t\}\times \supp X_t \right)\cap A\neq\emptyset\}. \] Arguments similar as those of \cite{dlg:wtsbmbs} yield that $\tau_A$ is a stopping time for the natural filtration of $X$ completed the usual way. Thus we have $\PP_{r,x}(\tau_A=r)=1$ or $0$. \label{com:ta} Following Dynkin~\cite[section II-6]{d:spde}, we say a point $(r,x)\in \R\times \R^d$ is $G$-regular for $A^{c}$ if $\PP_{r,x}$-a.s. $\tau_A=r$. Let $A^{Gr}$ denote the set of all points that are G-regular for $A^{c}$. From the known path properties of super-Brownian motion it is obvious that int$(A)\subset A^{Gr}\subset \bar A$, where int$(A)$ denotes the interior of $A$. We set $T_A=\inf \left\{s>0, (\zeta_s,\hat W_s) \in A\right\}$. Following \cite{dlg:wtsbmbs} it is easy to deduce from Theorem~\ref{th:ucap} the next result. \begin{proposition} \label{prop:sreg} Let $A\in \cb(\R\times \R^d)$. The following properties are equivalent: \begin{enumerate} \item $(r,x)$ is $G$-regular for $A^{c}$; \item $\N_{r,x}\left[\cg^*\cap A\neq\emptyset \right]=\infty $; \item $\ce_{\mbox{\rm{$\textbf{x}_r$}}}$-a.s. $T_A=0$; \item $\capa (A_{r,x}\cap E)=\infty $, where $A_{r,x}=\{(s,y); (s+r,y+x)\in A\}$. \end{enumerate} \end{proposition} We can give a straightforward analytic consequence of Proposition~\ref{prop:sreg} and the link between super-Brownian motion and nonlinear differential equation. \begin{corollary} Let $Q$ a domain in $E$ such that $(0,0)\in \partial Q$. The following three conditions are equivalent. \begin{enumerate} \item $(0,0)$ is $G$-regular for $Q$ ; \item $\capa (Q^c\cap E)=\infty$ ; \item There exists a nonnegative solution of $\frac{\partial u}{\partial t} +\frac{\Delta}{2} u=2 u^2 $ in $Q$ such that \[ \lim_{(s,y)\rightarrow (0,0),\; (s,y)\in Q} u(s,y)=\infty. \] \end{enumerate} \end{corollary} The equivalence of assertions 1) and 3) is due to Dynkin~\cite[Theorem II.6.1]{d:spde}. The equivalence of 1) and 2) is given by Proposition~\ref{prop:sreg}. Finally, using Theorem~\ref{th:ucap} we give in section 5 an estimation of the hitting probability of the support of $X_1$. And we prove that in dimension $d\geq 2$, the support of super-Brownian motion and the range of $d$-dimensional Brownian motion are intersection-equivalent. \section{Equivalence of capacities for a weighted Sobolev space} \label{sec:sobolev} \noindent In this section, we introduce a new capacity, associated with a weighted Sobolev space, which is equivalent to the capacity $\capa$. This capacity will be very useful in the next section to prove the upper bound for Theorem~\ref{th:ucap}. If $S$ is an open subset of $\R^p$, we denote by $C_0^\infty (S)$ the set of all functions of class $C^\infty $ defined on $S$ with compact support. If $f$ is a measurable function defined on $S$ then $\norm{f}_\infty =\sup_{x\in S} \val{f(s)}$. We consider the Hilbert space $L^2(p)=\left\{f \in \cb(E);\; \norm{f}_{(p)}<\infty \right\}$, where $\norm{f}_{(p)}^2=\iint_E dtdx\; p(t,x) f(t,x)^2$. Notice the kernel defined on $E\times E$ by $k(t,x;s,y)=p(t-s,x-y)p(t,x)^{-1}$ is nonnegative and lower semi-continuous. Thus we can introduce the operator $\Lambda$ defined on the set of nonnegative functions $f\in \mathcal{B}(E)$ by: \[ \Lambda(f)=p^{-1} [p*(pf)]=\iint_E dsdy\; k(\cdot,\cdot;s,y)p(s,y) f(s,y), \] where $*$ denotes the usual convolution product on $E$. Furthermore, the function $\Lambda(f)$ is even lower semi-continuous (see \cite[Lemma 2.2.1]{f:tplcs}). We define the capacity $\Capa$ on $E$ in the following way: if $A\subset E$, then \[ \Capa (A)=\inf \left\{ \norm{f}_{(p)}^2; \;f\geq 0,\; f\in L^2(p),\; \Lambda(f)\geq 1 \quad\text{on} \quad A \right\}, \] with the convention $\inf \emptyset=\infty $. Notice this capacity is not invariant by translation in time or space. This capacity is an outer capacity (see Meyers \cite[Theorem 1]{m:tcpf}). Moreover, it coincides with the capacity $\capa$ on the analytic sets (see \cite[Theorem 14]{m:tcpf}). Now, we want to connect this capacity to an analytic capacity (see Baras and Pierre \cite{bp:ppsdm} for similar results but with different norms). Therefore we consider the weighted Sobolev space $W_D$ which is the completion of $C^\infty_0(E)$ with respect to the norm $\norm{\cdot}_D$, defined by \[ \norm{\varphi}_D^2=\norm{\partial_t \varphi}_{(p)}^2+ \sum_{i=1}^d \norm{\partial_i (\log p) \; \partial_i \varphi }_{(p)}^2+ \sum_{i=1}^d \norm{\partial^2_{ii}\varphi}_{(p)}^2,\quad \varphi\in {C}_{0}^{\infty }(E), \] with the usual notations $\partial_t g(t,x)=\frac{\partial g}{\partial t} (t,x)$, $\partial_i g(t, x)=\frac{\partial g}{\partial x_i} (t,x)$ for $x=(x_1,\cdots,x_d)\in \R^d$ and $\partial^2_{ii}=\partial_i\partial_i$. Notice the non zero constants do not belong to $W_D$. We can introduce the outer capacity $\capaD$ associated to $W_D$ defined as follows. For any compact set $K\subset E$, we set \begin{align*} \capaD (K) &=\inf \left\{\norm{\varphi}_D^2; \;\varphi\in C_0^\infty (E), \;\varphi\geq 0, \;\varphi\geq 1 \text{ on } K \right\}\\ &=\inf \left\{\norm{\varphi}_D^2; \;\varphi\in C_0^\infty (E), \;\varphi\geq 0, \;\varphi\geq 1 \text{ on a neighborhood of } K \right\}. \end{align*} Then we set for any open set $G\subset E$, \begin{equation} \label{eq:capdGK} \capaD (G)=\sup\left\{\capaD(K); \;K\subset G, \;K \quad\text{compact} \right\}, \end{equation} and, for any analytic set $A\subset E$, \[ \capaD (A)=\inf\left\{\capaD(G); A\subset G, \;G \quad\text{open} \right\}. \] Notice the definition is consistent (see \cite{ah:fspt} for example). \begin{proposition} There exists a constant $C$ such that for any set $A\subset E$, \[ \Capa (A)\leq \capaD(A)\leq C \Capa(A). \] \end{proposition} \noindent \textbf{Proof.} Since the two capacities are outer capacity, it is enough to consider open sets. Now, using \reff{eq:capdGK} and \cite[Theorem 8]{m:tcpf}, we see it is enough to consider compact sets. Let us introduce the operator $H={\partial}_t-\inv{2}{\Delta}$. We consider a non empty compact set $K\subset E$. Let $\varphi\in C_0^\infty(E) $ be such that $\varphi\geq 0$ and $\varphi\geq 1$ on $K$. Notice that (in the distribution sense) $ H p=\delta_{(0,0)}$, where $\delta_{(0,0)}$ is the Dirac mass at ${(0,0)}\in {\mathbb R}\times\R^{d}$. Then we have $p* [H (p\varphi)]= (H p)*(p\varphi)= p\varphi$. The function $f=p^{-1} \val{H(p\varphi)}=\val{H(\varphi)-(\nabla \log p, \nabla \varphi)}$ is nonnegative and $\Lambda(f)\geq \varphi$. Thus we have $\Lambda(f)\geq 1$ on $K$. We also have $\norm{f}_{(p)}\leq \norm{\varphi}_D$. Hence we have $\Capa (K)\leq \norm{\varphi}_D^2$. The first inequality follows by taking a sequence $(\varphi_n)$ such that $\norm {\varphi_n}_D^2$ converges to $\capaD (K)$. To prove the other inequality, let us consider a nonnegative function $f_1\in L^2(p)$, such that $\Lambda (f_1)\geq 1$ on $K$. Notice it implies $\norm{f_1}_{(p)}>0$. Let $\delta >0$. It is easy to construct a function $\varepsilon\in L^2(p)$ such that $\varepsilon >0$ on $E$ and $\norm{\varepsilon}_{(p)}\leq \delta\norm{f_1}_{(p)}$. We set $f_2=f_1+\varepsilon$. Since the function $\Lambda (f_2)$ is lower semi-continuous, the set $\{(t,x)\in E; \;\Lambda (f_2)>1\}$ is open and it also contains $K$. It is then obvious that for $\delta'>0$ small enough, if we set $f_3(t,x)=f_2(t,x)\ind_{\{\delta'< t<{\delta'}^{-1}, \val{x}<{\delta'}^{-1} \}}$ for $(t,x)\in E$, we get $\Lambda (f_3)> 1$ on an open set containing $K$. Let us introduce a nonnegative function $h\in C_0^\infty (E)$ such that $\iint_E h(t,x)dtdx=1$. For $\theta>0$, we write $h_\theta(t,x)=\theta^{-d-1} h(t/\theta,x/\theta)$. Now using the uniform continuity of $p$ on $[\delta'/2,\infty )\times \R^d$, it is easy to see that if $f=h_\theta *f_3$ , then $\Lambda (f)> 1$ on an open set containing $K$ for $\theta$ small enough. The function $f$ is nonnegative, belongs to $C_0^\infty(E) $ and the function $\Lambda(f)$ is of class $C^\infty $. We can choose $\delta$ and $\theta$ small enough so that $\norm{f}_{(p)}\leq 2\norm{f_1}_{(p)}$. \label{com:convol} Let $\alpha \in C_0^\infty ([0,\infty ))$ such that $0\leq \alpha\leq 1$, $\alpha=1$ on $[0,1/2]$ and $\alpha=0$ on $[1,\infty) $. Let $\xi \in C_0^\infty (\R^d)$ such that $0\leq \xi\leq 1$ and $\xi=1$ in a neighborhood of $0$. We define $\alpha_n(t)=\alpha (t/n)$ and $\xi_n(x)=\xi(x/n)$. The function $\varphi_{n}= \alpha_n \xi_n \Lambda(f)$ belongs to $C_0^\infty (E)$, is nonnegative and $\varphi_{n} \geq 1$ on a neighborhood of $K$ for $n$ great enough. Let us now give two key lemmata. If $M$ is a bounded operator from $L^2(p)$ into itself, we denotes by $\normt{M}_{(p)}= \sup\{\norm{M(f)}_{(p)}; f\in L^2(p),\; \norm{f}_{(p)}=1\}$ its norm. We define the operator $\Lambda_0$: for $f\in \cb(E)$ nonnegative, $\Lambda_0(f)(t,x)=t^{-1} \Lambda(f)(t,x)$, $(t,x)\in E$. For $T>0$, let us introduce $E_T=(0, T)\times \R^d$. \begin{lemma} \label{lem:opL} The operators $\ind_{E_T}\Lambda$ and $\Lambda_0$ are bounded operators from $L^2(p)$ into itself. Furthermore, we have $\normt{\ind_{E_T}\Lambda}_{(p)}\leq T/\sqrt{2}$ and $\normt{\Lambda_0}_{(p)}\leq {2}$. \end{lemma} \noindent \textbf{Proof} of Lemma \ref{lem:opL}. Let $f\in L^2(p)$. The Cauchy-Schwarz inequality implies: \begin{align*} \norm{\Lambda_0(f) }_{(p)}^2&= \iint_{E} dt dx\; t^{-2} p(t,x)^{-1} \left[\iint_E ds dy\; p(t-s,x-y) p(s,y) f(s,y) \right]^2\\ &\leq \iint_{E} dt dx\; t^{-2}p(t,x)^{-1} \iint_E ds' dy'\; p(t-s',x-y') p(s',y'){s'}^{-1/2}\ind_{s'\leq t} \\ &\phantom{\leq \iint_{E_T} dt dx\; t^{-2}p(t,x)^{-1} } \iint_E ds dy\; p(t-s,x-y) p(s,y) {s}^{1/2}f(s,y)^2 \\ & =2\iint_{E} dt dx\; t^{-2} t^{1/2}\iint_E ds dy\; p(t-s,x-y) p(s,y) {s}^{1/2}f(s,y)^2 \\ & \leq 2 \iint_{E} ds dy\; p(s,y) f(s,y)^2 {s}^{1/2}\int_s^\infty t^{-3/2}dt \leq 4\norm{f}^2_{(p)}. \end{align*} Hence the operator $\Lambda_0$ is a bounded operator from $L^2(p)$ into itself. And we have $\normt{\Lambda_0}_{(p)}\leq {2}$. The operator $\ind_{E_T}\Lambda$ can handled in a very similar way. \findemo %%%%%%%%%%%% %Let us fix $T>0$ and $f\in C_0^\infty (E)$. Using Cauchy-Schwarz inequality, we get that %\begin{align*} %\norm{\ind_{E_T} \Lambda(f)}_{(p)}^2 %&=\iint_{E_T} dt dx\; p(t,x)^{-1} \left[\iint_E ds dy\; p(t-s,x-y) p(s,y) f(s,y) \right]^2\\ %&\leq \iint_{E_T}dt dx\; p(t,x)^{-1} \iint_E ds' dy'\; p(t-s',x-y') p(s',y') \\ %&\hspace{4cm} \iint_E ds dy\; p(t-s,x-y) p(s,y) f(s,y)^2 \\ %& =\iint_{E_T} dt dx\; t\iint_E ds dy\; p(t-s,x-y) p(s,y) f(s,y)^2 \\ %&\leq \frac{T^2}{2} \norm{f}_{(p)}^2. %\end{align*} %Hence we get that $\norm{\ind_{E_T}\Lambda (f)}_{(p)}\leq \frac{T}{\sqrt{2}} \norm{f}_{(p)}$. The lemma then follows from classical arguments using the density of $C^\infty _0(E)$ in $L^2(p)$. \begin{lemma} \label{lem:opL1} The operators defined on $C_0^\infty (E) $ by: $g \in C_0^\infty(E) $ \begin{align*} \Lambda_1(g)& ={\partial_t} \Lambda(g),\\ \text{for}\quad i\in \{1,\cdots,d\}, \quad \Lambda_{2,i}(g)& =\inv{2}\partial^2_{ii} \Lambda(g),\\ \text{ and for}\quad i\in \{1,\cdots,d\}, \quad \Lambda_{3,i}(g)& =\partial_{i}( \log p) \;\partial_{i}\Lambda(g), %\\\text{and}\quad \Lambda_0(g)(t,x)&={t}^{-1} \Lambda(g)(t,x)\quad\text{for}\quad (t,x)\in E, \end{align*} can be uniquely extended into bounded operators from $L^2(p)$ into itself. And we have \begin{align} \label{eq:opL1} \normt{\Lambda_1}_{(p)}& \leq 1+3d,\\ \label{eq:opL2i} \text{for}\quad i\in \{1,\cdots,d\}, \quad \normt{\Lambda_{2,i}}_{(p)}& \leq 1,\\ \label{eq:opL3} \text{for}\quad i\in \{1,\cdots,d\}, \quad \normt{\Lambda_{3,i}}_{(p)}& \leq 4. %\label{eq:opL4} \normt{\Lambda_0}_{(p)}& \leq 2. \end{align} \end{lemma} \noindent The proof of this lemma is given in appendix. We now bound $\norm{\varphi_{n}}_D$. Lemma \ref{lem:opL} provides an upper bound for $\norm{\partial_t \varphi_{n}}_{(p)}$: \begin{align} \nonumber \norm{\partial_t \varphi_{n}}_{(p)} & \leq \norm{\partial_t \alpha_n}_\infty \norm{\ind_{E_n}\Lambda(f)}_{(p)}+ \norm{\Lambda_1{f}}_{(p)}\\ \label{eq:dtphin} & \leq \left(\norm{\partial_t \alpha}_\infty 2^{-1/2} +\normt{\Lambda_1}_{(p)}\right)\norm{f}_{(p)}. \end{align} \noindent Using Lemma \ref{lem:opL} we derive an upper bound for $\sum_{i=1}^d \norm{\partial_{i} \log p \;\partial_{i}\varphi_{n}}_{(p)}$: \begin{align} \nonumber \sum_{i=1}^d \norm{\partial_{i} \log p \;\partial_{i}\varphi_{n}}_{(p)} &\leq \sum_{i=1}^d\left(\norm{\Lambda_{3,i}(f)}_{(p)}+\sup_{x\in \R^d} \val{x_i \partial_i \xi(x)}\norm{\Lambda_0(f)}_{(p)}\right)\\ \label{eq:dilogpdiphin} & \leq \sum_{i=1}^d \left(\normt{\Lambda_{3,i}}_{(p)}+\sup_{x\in \R^d} \val{x_i \partial_i \xi(x)}\normt{\Lambda_0}_{(p)} \right)\norm{f}_{(p)}. \end{align} In order to give an upper bound for $\sum_{i=1}^d \norm{\partial^2_{ii} \varphi_{ n}}_{(p)}$, we need an intermediary lemma. \begin{lemma} \label{lem:oPL2} There exists a constant $c_1$ (depending on $\xi$) such that for all $n\geq 1$, $g\in C_0^\infty (E)$, $i\in\{1,\cdots,d\}$, \begin{equation*} % \label{eq:intermed} \norm{\ind_{E_n} \partial_i \xi_n \;\partial_i \Lambda(g) }_{(p)}\leq c_1 n^{-1/2}\norm{g}_{(p)}. \end{equation*} \end{lemma} \noindent \textbf{Proof.} Recall that $\xi_n$ has compact support. Then, an integration by parts, Cauchy-Schwarz inequalities and Lemma~\ref{lem:opL} give for $1\leq i\leq d$, \begin{align*} \norm{\ind_{E_n} \partial_i \xi_n \partial_i \Lambda(g)}_{(p)}^2 = &-\iint_E p \ind_{E_n} \Lambda(g)(\partial_i \xi_n )^2\partial^2_{ii} \Lambda(g)\\ &\hspace{1cm}- \iint_{E} p\ind_{E_n} \Lambda(g) (\partial_i \xi_n)^2 \partial_i \Lambda(g) \partial_i \log p\\ &\hspace{1cm}- 2\iint_{E} p\ind_{E_n} \Lambda(g) \partial_i \xi_n \partial_i \Lambda(g) \partial^2_{ii} \xi_n\\ \leq & \norm{\partial_i\xi_n}_\infty ^2\norm{\ind_{E_n} \Lambda(g)}_{(p)}\norm{\partial^2_{ii} \Lambda(g)}_{(p)}\\ &\hspace{1cm}+ \norm{\partial_i\xi_n}_\infty ^2\norm{\ind_{E_n} \Lambda(g)}_{(p)}\norm{\partial_i \Lambda(g)\;\partial_i \log p}_{(p)}\\ &\hspace{1cm}+2\norm{\partial^2_{ii} \xi_n}_\infty \norm{\ind_{E_n} \Lambda(g)}_{(p)}\norm{\ind_{E_n} \partial_i \xi_n \partial_i \Lambda(g)}_{(p)}\\ \leq &2^{-1/2} n^{-1}\left[2\normt{\Lambda_{2,i}}_{(p)} +\normt{\Lambda_{3,i}}_{(p)}\right]\norm{\partial_i \xi}_\infty ^2\norm{g}_{(p)}^2\\ &\hspace{1cm}+2^{1/2} n^{-1}\norm{\partial^2_{ii} \xi}_\infty \norm{g}_{(p)}\norm{\ind_{E_n} \partial_i \xi_n \partial_i \Lambda(g)}_{(p)}. \end{align*} Notice that if $a,b,c$ are positive then $a^2\leq c^2+ba$ implies $a\leq c+b$. Thus we get \begin{multline*} \norm{\ind_{E_n} \partial_i \xi_n \partial_i \Lambda(g)}_{(p)}\\ \leq 2^{-1/4}n^{-1/2}\left[2\normt{\Lambda_{2,i}}_{(p)} +\normt{\Lambda_{3,i}}_{(p)}\right]^{1/2}\norm{\partial_i \xi}_\infty \norm{g}_{(p)}+ 2^{1/2} n^{-1}\norm{\partial _{i,i}^2 \xi}_\infty \norm{g}_{(p)}, \end{multline*} which ends the proof. \findemo Using this lemma and Lemma \ref{lem:opL}, we get that \begin{align} \nonumber \sum_{i=1}^d \norm{\partial^2_{ii} \varphi_{ n}}_{p} &\leq \sum_{i=1}^d\left[2\norm{\Lambda_{2,i}(f)}_{(p)}+\norm{\partial^2_{ii} \xi_n}_\infty \norm{\ind_{E_n} \Lambda(f)}_{(p)}+2\norm{\ind_{E_n}\partial_{i}\xi_n\; \partial_{i}\Lambda(f))}_{(p)}\right]\\ \label{eq:diiphin} &\leq \sum_{i=1}^d \left[2\normt{\Lambda_{2,i}}_{(p)}+2^{-1/2} n^{-1}\norm{\partial^2_{ii} \xi}_\infty +2c_1n^{-1/2}\right]\norm{f}_{(p)}. \end{align} \noindent Then we deduce from \reff{eq:dtphin}, \reff{eq:dilogpdiphin}, \reff{eq:diiphin} and Lemma~\ref{lem:opL1} that there exists a constant $c_2$ independent of $f$ and $n\geq 1$ such that \[ \norm{\varphi_{n}}_D\leq c_2 \norm{f}_{(p)}. \] Thus we have $\norm{\varphi_n}_D\leq 2 c_2 \norm{f_1}_{(p)}$. The second inequality of the proposition is then obvious with $C=4\,c_2^2$. \findemo We shall need the following lemma. \begin{lemma} \label{lem:phiK} There exists a constant $\gamma$ such that if $K\subset E$ is a compact set such that $\capaD (K)>0$, there exists $\varphi\in C_0^\infty (E)$ which satisfies: 1) $0\leq \varphi\leq 1$, 2) $\varphi=1$ on a neighborhood of $K$, 3) $\norm{\varphi}_D^2\leq \gamma \capaD (K)$. \end{lemma} \noindent The proof is classic, but we give it for completeness. \noindent \textbf{Proof.} Let $h\in C^\infty ([0,\infty ))$ such that $0\leq h\leq 1$, $h=0$ on $[0,1/4]$ and $h=1$ on $[3/4,\infty )$. Since $\capaD (K)>0$, there exists $g \in C_0^\infty (E)$ such that $g\geq 0$, $g\geq 1$ in a neighborhood of $K$, and $2\capaD (K)\geq \norm{g}_D^2$. Let $\varphi= h\circ g$. The function $\varphi\in C_0^\infty (E)$ satisfies 1) and 2). Let us check 3). We have \begin{align*} \norm{\partial_t \varphi}_{(p)} &\leq \norm{h'}_\infty \norm{\partial_t g}_{(p)},\\ \norm{\partial_{i} \log p \;\partial_{i}\varphi}_{(p)} &\leq \norm{h'}_\infty \norm{\partial_{i} \log p \;\partial_{i}g}_{(p)}\\ \norm{ \partial^2_{ii} \varphi}_{(p)} &\leq \norm{h'}_\infty \norm{\partial^2_{ii} g}_{(p)} +\norm{(h''\circ g) \;(\partial_i g)^2}_{(p)}. \end{align*} Only the upper bound for the second right hand-side term of the last inequality is not obvious. We first search an upper bound for $\norm{(\partial_i \varphi_1)^2/(1+\varphi_1)}_{(p)}$, where $\varphi_1\in C^\infty _0(E)$ is a nonnegative function. An integration by parts and Cauchy-Schwarz inequality give \begin{align*} \iint_E p\frac{(\partial_i \varphi_1)^4}{(1+\varphi_1)^2} &= 3 \iint_E p\; \partial_{ii}^2 \varphi_1 \frac{(\partial_i \varphi_1)^2}{1+\varphi_1} +\iint_E p \;\partial_i \log p \;\partial_i \varphi_1 \frac{(\partial_i \varphi_1)^2}{1+\varphi_1}\\ &\leq (3 \norm{ \partial^2_{ii} \varphi_1}_{(p)} +\norm{\partial_{i} \log p \;\partial_{i}\varphi_1}_{(p)})\norm{{(\partial_i \varphi_1)^2}/{(1+\varphi_1)}}_{(p)}. \end{align*} Thus we get \begin{equation} \label{eq:maj-normediphi} \norm{(\partial_i \varphi_1)^2/(1+\varphi_1)}_{(p)}\leq 3 \left(\norm{ \partial^2_{ii} \varphi_1}_{(p)}+ \norm{\partial_{i} \log p \;\partial_{i}\varphi_1}_{(p)}\right). \end{equation} Since we have $\val{h''(t)}\leq 2(1+t)^{-1} \norm{h''}_\infty $, taking $\varphi_1=g$ in the above inequality we deduce that \begin{align*} \norm{(h''\circ g) \;(\partial_i g)^2 }_{(p)}& \leq 2\norm{(\partial_i g)^2/(1+g)}_{(p)}\norm{h''}_{\infty }\\ & \leq 6(\norm{ \partial^2_{ii} g}_{(p)}+ \norm{\partial_{i} \log p \;\partial_{i}g}_{(p)})\norm{h''}_{\infty }. \end{align*} The previous inequalities imply there exists a constant $c$ depending only on $h$ and $d$ such that $\norm{\varphi}_D\leq c \norm{g}_D$. Thus 3) holds with $\gamma=2c^2$. \findemo \section{Upper bound for hitting probabilities} \label{sec:upper} \noindent In this section we prove the second inequality of Theorem \ref{th:ucap} for compact sets. Let us introduce $K\subset E_T$ a compact set such that $\capaD (K)>0$. Let $\varphi$ be as in Lemma \ref{lem:phiK}. We set $\varphi=0$ outside $E$. We introduce the function $\psi=1-\varphi$, which takes values in $[0,1]$. We consider the function $u$ defined on $\R\times \R^d$ by $u(t,x)=\N_{t,x}[\cg^*\cap K\neq\emptyset]$ ($\in [0,\infty ]$). With the convention $0.\infty =0$, the function $u\psi^4$ is bounded nonnegative and of class $C^\infty $ on $\R\times \R^d$. Let $(B_t,t\geq 0)$ denote under $\rP_0$ a $d$-dimensional Brownian motion started from 0. It\^o's formula implies that for all $t\geq 0$, $\rP_0$-a.s., \begin{multline*} u\psi^4(t,B_t)=u\psi^4(0,0)+\int_0^t {\partial_t (u\psi^4)}(s,B_s) ds \\ +\int_0^t \frac{\Delta}{2} (u\psi^4)(s,B_s) ds +\int_0^t \nabla (u\psi^4)(s,B_s) dB_s. \end{multline*} Consider the stopping time $T_a=T\wedge \inf\{t>0; \val{B_t}\geq a\}$. We can then apply the stopping Theorem at time $T_a$ and get \begin{align*} \rE_0 u\psi^4(T_a,B_{T_a}) &=u(0,0) +\rE_0 \int_0^{T_a} {\partial_t (u\psi^4)}(s,B_s) ds +\rE_0\int_0^{T_a} \frac{\Delta}{2} (u\psi^4)(s,B_s) ds\\ &=u(0,0) +\rE_0 \int_0^{T_a} \bigg [2u^2 \psi^4+4u\psi^3{\partial_t \psi}+4 (\nabla u,\nabla \psi)\psi^2 \\ &\phantom{=u(0,0)+\E_0 \int_0^{T_a} [.]} +6 u \psi^2 (\nabla \psi, \nabla \psi)+2 u \psi^3\Delta\psi \bigg](s,B_s) ds.\\ \end{align*} We have used that $\partial_t u+\inv{2} \Delta u =2u^2$ to get the last equality. Notice that each integrand is either nonnegative or bounded. By dominated convergence and monotone convergence, we get as $a$ goes to infinity \begin{align*} u(0,0)+2\norm{u\psi^2\,\ind_{E_T}}_{(p)}^2=\rE_0 u\psi^4(T,B_{T}) - \iint_{E_T} p\bigg[ &4u\psi^3{\partial_t \psi}+4 (\nabla u,\nabla \psi)\psi^2 \\ &+6 u \psi^2 (\nabla \psi, \nabla \psi)+2 u \psi^3\Delta\psi \bigg]. \end{align*} Since $K\subset E_T$, we deduce that $u(t,x)=0$ for $t\geq T$. Thus we have: \begin{align} \nonumber u(0,0)+2\norm{u\psi^2}_{(p)}^2= - \iint_{E} p\bigg[ &4u\psi^3{\partial_t \psi}+4 (\nabla u,\nabla \psi)\psi^2 \\ \label{eq:egalu00} &+6 u \psi^2 (\nabla \psi, \nabla \psi)+2 u \psi^3\Delta\psi \bigg]. \end{align} We now bound the right hand side. Using Cauchy-Schwarz inequality, that $0\leq \psi\leq 1$, and that $-\varphi$ and $\psi$ have the same derivatives, we get \begin{align*} -\iint_E pu\psi^3 {\partial_t \psi} &\leq \norm{u\psi^2}_{(p)} \norm{{\partial_t \varphi }}_{(p)},\\ -\iint_E pu \psi^3 \partial^2_{ii} \psi &\leq \norm{u\psi^2}_{(p)} \norm{\partial^2_{ii}\varphi}_{(p)}, \end{align*} \begin{align*} \text{and}\quad -\iint_E p u\psi^2 (\nabla \psi, \nabla \psi) & \leq \norm{u\psi^2}_{(p)} \sum_{i=1}^d \norm{(\partial_i\varphi)^2}_{(p)}\\ & \leq 2\norm{u\psi^2}_{(p)} \sum_{i=1}^d \norm{{(\partial_i\varphi)^2}/{(1+\varphi)}}_{(p)}\\ & \leq 6\norm{u\psi^2}_{(p)} \sum_{i=1}^d (\norm{ \partial^2_{ii} \varphi}_{(p)}+ \norm{\partial_{i} \log p \;\partial_{i}\varphi}_{(p)}),\\ \end{align*} where we have used \reff{eq:maj-normediphi} with $\varphi_1=\varphi$ for the last inequality. Now an integration by parts and Cauchy-Schwarz inequality give \begin{align*} -\iint_E p \psi^3 (\nabla u, \nabla \psi) &= \iint_E pu \psi^2\left[\psi (\nabla \log p, \nabla \psi)+3 (\nabla \psi, \nabla \psi)+\psi \Delta\psi\right]\\ &\leq \norm{u\psi^2}_{(p)} \sum_{i=1}^d \left[ \norm{\partial_{i} \log p \;\partial_{i}\varphi}_{(p)} +3\norm{ (\partial_{i} \varphi)^2}_{(p)}+\norm{ \partial^2_{ii} \varphi}_{(p)} \right]\\ &\leq 19 \norm{u\psi^2}_{(p)} \sum_{i=1}^d \left[ \norm{\partial_{i} \log p \;\partial_{i}\varphi}_{(p)} +\norm{ \partial^2_{ii} \varphi}_{(p)} \right], \end{align*} where we have used again \reff{eq:maj-normediphi} for the last inequality. Taking those results together, we deduce from \reff{eq:egalu00} that \[ u(0,0)+2\norm{u\psi^2}_{(p)}^2\leq c_3 \norm{u\psi^2}_{(p)}\norm{\varphi}_D, \] where the constant $c_3$ depends only on $d$. Since $\norm{u\psi^2}_{(p)}$ is finite (recall $u\psi$ is bounded, and zero on $[T,\infty )\times \R^d$), this implies that $\norm{u\psi^2}_{(p)}\leq c_3 \norm{\varphi}_D$ and hence $u(0,0)\leq c_3^2\norm{\varphi}_D^2$. This last inequality and the definition of $\varphi$ imply that \[ \N_{0,0}[\cg^*\cap K\neq\emptyset]=u(0,0)\leq c_3^2\gamma \capaD(K)\leq c_3^2 \gamma C \Capa (K) = c_3^2 \gamma C \capa (K). \] \section{Lower bound for hitting probabilities and proof of Theorem \ref{th:ucap}} \label{sec:lower} \noindent In this section, we prove the first inequality of Theorem \ref{th:ucap} for compact sets. Let us introduce a compact set $K\subset E$, $\nu$ a probability measure on $K$, and $T>0$ such that $K\subset E_T$. We consider the probability measure $\mu$ defined on $\cw_{0,0}$ by \[ \mu(d\rw) =\iint_E \nu(dt,dx) \; {\rm P}_0^{t,x}(d\rw), \] where ${\rP}_0^{t,x}$ is the law on $\cw_{0,0}$ of the Brownian bridge starting at time $0$ at point $0$ and ending at time $t$ at point $x$. Notice that the measure $\mu$ is in fact a measure on $\cw_{0,0}^*$, the set of non trivial path in $\cw_{0,0}$ (a trivial path is a path of lifetime zero). The measure ${\rP}_0^{t,x}$ can also be viewed as a probability measure on the canonical space $C(\R^+,\R^d)$ endowed with the filtration $(\ccc_t)$ generated by the coordinate mappings. Let $\rP_0$ be the law on the canonical space of the standard Brownian motion. For $s\in [0,t)$, we have \[ {\rm P}_0^{t,x}(d\rw)_{\mid \ccc_s}=\frac{p(t-s,x-\rw(s))}{p(t,x)} {\rm P}_0(d\rw)_{\mid \ccc_s}. \] We consider the energy of $\mu$ with respect to the process $(W_s)$ (see \cite{lg:hppt} for a precise description and definition). Thanks to \cite[Proposition 1.1]{lg:hppt} we have: \[ \ce(\mu) =2 \int_0^\infty ds\;\rP_0 \left[\left(\iint_E \nu(dt,dx) p(t-s,x-\rw(s))/p(t,x)\right)^2\right]=2I(\nu). \] Now, using \cite[Proposition 5]{dlg:wtsbmbs}, we know there exists an additive functional $A$ of the Brownian snake killed when its lifetime reaches $0$ such that: \begin{itemize} \item[(i)] For every Borel function $F\geq 0$ on $\cw_{0,0}^*$, $\N_{0,0}\left[\int_0^\infty F(W_s)dA_s\right] =\int\mu(d\rw)F(\rw) $. \item[(ii)] $\N_{0,0}[A_\infty ^2]=2\ce(\mu)$. \end{itemize} We deduce from (i) that the additive functional increases only when $\hat W_s\in \supp \nu\subset K$. Therefore, using the Cauchy-Schwarz inequality, we get \[ \N_{0,0}[\cg^*\cap K\neq\emptyset] \geq \N_{0,0}[A_\infty >0] \geq \N_0[A_\infty ]^2/\N_0[A_\infty ^2]. \] We get $\N_{0,0}[\cg^*\cap K\neq\emptyset] \geq [4I(\nu)]^{-1}$. Since the above inequality is true for any probability $\nu$ on $K$, we get that \[ \N_{0,0}[\cg^*\cap K\neq\emptyset]\geq 4^{-1} \Capa (K)= 4^{-1} \capa (K). \] \noindent \textbf{Proof} of Theorem \ref{th:ucap}. Notice the application defined on $\cb(E)$ by $T(A)=\N_{0,0}[\cg^*\cap A\neq \emptyset]$ for $A\in \cb(E)$ is a Choquet capacity (see~\cite[th\'eor\`eme 1]{c:fatc}). Since the capacity $\capa$ is an inner capacity (see \cite[Theorem 12]{m:tcpf}), it is enough to prove the theorem for compact subsets of $E$. The result is then given by the previous section (with $C_0=c_3^2 \gamma C$) and the above result. \findemo \section{Brownian range and support of $X_1$} In this section, we first give an estimation for the hitting probabilities of the support of $X_1$. Then we prove that the range of Brownian motion and the support of super-Brownian motion at fixed time are intersection equivalent. Let us fix $d\geq 2$. We denote by $\capa_{d-2}$ the usual Newtonian (logarithmic if $d=2$) capacity in $\R^d$: \[ \capa_{d-2} (A)=\left[\inf \iint_{\R^d\times \R^d}\rho(dx)\rho(dy)\;h_{d-2}(\val{x-y})\right]^{-1}, \] with $h_{\gamma}(r)=r^{-\gamma}$ if $\gamma>0$ and $h_0(r)=\log_+(1/r)$. The infimum is taken over all probability measures $\rho$ on $\R^d$ such that $\rho(A)=1$. Let $B(0,h)$ be the open ball of ${\mathbb R}^d$ centered at $0$ with radius $h$. \begin{proposition} \label{prop:hitting} Let $M>0$. There exist two positive constants $a$ and $b$ such that for any Borel set $A\subset B(0,1)$, for any finite measure $\mu$ on $B(0,1)$, with $(\mu,\ind)\leq M$, we have \[ a (\mu,\ind) \capa_{d-2} (A)\leq \P_{0,\mu}[\supp X_1 \cap A \neq \emptyset]\leq b(\mu,\ind) \capa_{d-2} (A). \] \end{proposition} \noindent \textbf{Proof.} Let $A\subset B(0,2)$ be a Borel set. Let $\nu$ be a probability measure on $E$ such that $\nu(\{1\}\times A)=1$. Then we have $\nu=\delta_{\{1\}}\times \rho$, where $\rho$ is a probability measure on $\R^d$ such that $\rho(A)=1$. We get \begin{multline*} I(\nu)= \iint_{(0,1)\times \R^d} dsdy\; p(s,y)\\ \iint_{A\times A} \rho(dx)\rho(dx')p(1-s,x-y)p(1-s,x'-y) {p(1,x)}^{-1}{p(1,x')}^{-1}. \end{multline*} Since $x,x'$ are in $B(0,2)$ and since $s\in(0,1)$ it is easy to see there exist two positive constants $a_1$ and $b_1$ (independent of $A$ and $\rho$) such that \[ a_1 I(\nu)\leq \iint_{A\times A} \rho(dx)\rho(dx') \;h_{d-2}(\val{x-x'}) \leq b_1 I(\nu). \] %\label{com:demoIE} \noindent This implies that for any Borel set $A\subset B(0,2)$, \[ a_1 \capa_{d-2} (A)\leq \capa (\{1\}\times A)\leq b_1 \capa_{d-2} (A). \] Since the capacity $\capa_{d-2}$ is invariant by translation, we get that for any Borel set $A\subset B(0,1)$, for any $x\in B(0,1)$, \[ a_1 \capa_{d-2} (A)\leq \capa (\{1\}\times A_{x})\leq b_1 \capa_{d-2} (A), \] where $A_x=\left\{y; y-x\in A\right\}$. We deduce from Theorem \ref{th:ucap} that \[ 4^{-1} a_1 \capa_{d-2} (A)\leq \N_{0,x}[\cg^*\cap (\{1\}\times A)\neq \emptyset] \leq C_0 b_1 \capa_{d-2} (A). \] Since $X_1=\sum_{i\in I}Y_1(W^i)$, where $\sum_{i\in I} \delta_{W^i}$ is a Poisson measure on $C(\R^+,\cw_{0})$ with intensity $\int\mu(dx)\N_{0,x}[\cdot]$, we have \[ \P_{0,\mu}[\supp X_1 \cap A \neq \emptyset]=1-\expp{-\int \mu(dx)\N_{0,x}[\supp Y_1 \cap A \neq \emptyset]}. \] Notice that $\N_{0,x}$-a.e., $\{1\}\times (\supp Y_1 \cap A )=\cg^* \cap (\{1\}\times A )$. Since $(\mu,\ind)0$. There exist two positive constants $a$ and $b$ such that for any Borel set $A\subset B(0,1)$, for any absolutely continuous probability measure $\pi$ on $B(0,1)$ with density bounded by $M$, for any finite measure $\mu$ on $B(0,1)$, with $(\mu,\ind)\leq M$, we have \[ a (\mu,\ind) \rP_\pi[\range_B\cap A\neq\emptyset ]\leq \P_{0,\mu}[\supp X_1 \cap A \neq \emptyset]\leq b(\mu,\ind) \rP_\pi[\range_B\cap A\neq\emptyset ]. \] \end{corollary} \noindent \textbf{Proof}. This is a consequence of Proposition \ref{prop:hitting} and the fact that there exist two positive constants $a_2$ and $b_2$ such that for any Borel set $A\subset B(0,1)$, for any absolutely continuous probability measure $\pi$ on $B(0,1)$ with density bounded by $M$, \begin{equation*} % \label{eq:rangeB} a_2 \capa_{d-2} (A)\leq \rP_\pi[\range_B\cap A\neq\emptyset ]\leq b_2 \capa_{d-2} (A) \end{equation*} (see for example \cite[Proposition 3.2]{p:iebpbp} for $d\geq 3$ and \cite{p:csf} for $d=2$). \findemo \section{Appendix} %\newcommand {\ca}{\mathcal{A}} \noindent In this section, we give the proof of Lemma~\ref{lem:opL1}, which relies on the properties of the Hermite polynomials. We first recall the definition and some properties of those polynomials. \subsection{Hermite polynomials} \noindent For $n=(n_1,\cdots,n_d)\in\N^d$, we set $\val{n}=\sum_{i=1}^d n_i$, $n!=\prod_{i=1}^d n_i!$ and $\sum_{n\geq 0}=\sum_{i=1}^d\sum_{n_i=0}^\infty $. For $j\in \{1,\cdots,d\}$, let $\delta(j)$ be the element of $\N^d$ such that $\delta(j)_i=\delta_{i,j}$, the standard Kronecker symbol. If $z=(z_1,\cdots,z_d)$ is an element of $\R^d$, then we set $z^n=\prod_{i=1}^d z_i^{n_i}$. Let $(\cdot,\cdot)$ be the Euclidean product on $\R^d$. The function $\varphi(z)=\expp{-[\val{z}^2-2(z,x)]/2}$ is an entire function defined on $\R^d$. We have \begin{equation} \label{eq:defH} \expp{-[\val{z}^2-2(x,z)]/2}=\sum_{n\geq 0} \inv{n!} z^n He_n(x), \end{equation} where the $n$-th term $He_n(x)$ is a polynomial of $(x_1,\cdots,x_d)$ of degree $\val{n}$ called the $n$-th Hermite polynomial. Those polynomials can easily be expressed with the usual one dimensional Hermite polynomials $(He_k^{(1)}, k\in \N)$: $He_n(x)=\prod_{i=1}^d He_{n_i}^{(1)}(x_i)$, where $ x=(x_1,\cdots,x_d)$. Now, let us recall some basic properties of the polynomials $He_n$. The following recurrence formula can be deduced from (\ref{eq:defH}) by derivating w.r.t. $z_i$: for all $n\in \N^d$ such that $n_i>0$, \begin{equation} \label{eq:reccH} He_{n}(x)=x_i He_{n-\delta(i)}(x)-(n_i-1)He_{n-2\delta(i)}(x),\quad \forall x\in \R^d, \end{equation} where by convention $He_{n-k\delta(i)}=0$ if $n_i-k<0$. The derivative formula can be deduced from \reff{eq:defH} by derivating w.r.t. $x_i$: for all $n\in \N^d$, \begin{equation} \label{eq:diffH} \partial_i He_n=n_i He_{ n-\delta(i)}. \end{equation} We also recall the upper bound for $He_n$ (see \cite[22.14.17]{as:hmff}): there exists a universal constant $1-1$, \begin{equation*} % \label{eq:hardy} \int_0^\infty dt\; t^{-2-k}\left[\int_0^t s^{k/2} h(s)ds \right]^2 \leq \frac{4}{(k+1)^2}\int_0^\infty dt\;h(t)^2 \end{equation*} for the first inequality and \reff{eq:norme-de-f} for the second one. This means that $\Lambda_{2,i}$, defined on $\ca$, can be uniquely extended into a bounded operator $\tilde \Lambda_{2,i}$ from $L^2(p)$ into itself. The above inequality implies $\normt{\tilde\Lambda_{2,i}}_{(p)} \leq 1$. For $i\in\{1,\cdots,d\}$, we set $\Lambda_{4,i}=\Lambda_{3,i}+2\Lambda_{2,i}$. Using \reff{eq:diffH} and \reff{eq:reccH}, we deduce from \reff{eq:egLh} that \begin{multline} \label{eq:lambda5i} \Lambda_{4,i}(h_{n,g})(t,x)\\ \begin{aligned}[b] &=\left[{-(x_i/\sqrt{t})}\partial_i He_{n}(x/\sqrt{t})+n_i(n_i-1)He_{n-2\delta(i)}(x/\sqrt{t})\right]t^{-1-\val{n}/2}\int_0^tds\; g(s)\\ &=-n_i He_{n}(x/\sqrt{t})t^{-1-\val{n}/2}\int_0^tds\; g(s). \end{aligned} \end{multline} Arguing as above, we get for $f\in \ca$, %and mimicking the proof of~\reff{eq:opL2if}, we get \begin{align*} \norm{\Lambda_{4,i}(f)}_{(p)}^2 &= \sum_{n\geq 0} n! n_i^2 \int_0^\infty dt\; t^{-2-\val{n}}\left[\int_0^t ds\; s^{\val{n}/2}g_n(s)\right]^2\\ &\leq \sum_{n\geq 0} n! n_i^2 \frac{4}{(\val{n}+1)^2} \int_0^\infty dt\; g_n(t)^2\\ &\leq 4 \norm{f}_{(p)}^2.\\ \end{align*} Thus the operators $\Lambda_{4,i}$ and $\Lambda_{3,i}$, defined on $\ca$, can be uniquely extended in bounded operators $\tilde\Lambda_{4,i}$ and $\tilde\Lambda_{3,i}$ from $L^2(p)$ into itself. Furthermore we have $\normt{\tilde\Lambda_{4,i}}_{(p)}\leq 2$ and $\normt{\tilde\Lambda_{3,i}}_{(p)}\leq \normt{\tilde\Lambda_{4,i}}_{(p)}+2\normt{\tilde\Lambda_{2,i}}_{(p)}\leq 4 $. The proof concerning $\Lambda_1$ easily follows from the previous results. From \reff{eq:egLh}, we get \begin{multline*} \Lambda_{1}(h_{n,g})(t,x)\\ =h_{n,g}(t,x)-\inv{2}\left[\val{n}He_n(x/\sqrt{t}) + \sum_{i=1}^d \frac{x_i}{\sqrt{t}} \partial_i He_n(x/\sqrt{t})\right]t^{-1-\val{n}/2}\int_0^t ds\; g(s). \end{multline*} Then using \reff{eq:lambda2i} and \reff{eq:lambda5i}, we get \[ \Lambda_1(h_{n,g})=\left[I+\inv{2}\sum_{i=1}^d \left[\Lambda_{4,i}+\Lambda_{3,i}\right]\right](h_{n,g})=\left[I+\sum_{i=1}^d \left[\Lambda_{4,i}-\Lambda_{2,i}\right]\right](h_{n,g}). \] This means that $\Lambda_1=I+\sum_{i=1}^d \left[\Lambda_{4,i}-\Lambda_{2,i}\right]$ on $\ca$. Hence $\Lambda_1$ can be uniquely extended in a bounded operator $\tilde\Lambda_1$ from $L^2(p)$ into itself and $\tilde\Lambda_1=I+\sum_{i=1}^d \left[\tilde\Lambda_{4,i}-\tilde\Lambda_{2,i}\right]$. We deduce that $\normt{\tilde\Lambda_1}_{(p)} \leq 1+3d$. \findemo \textbf{Second step}. We first consider the operators $\Lambda_{3,i}$ for $i\in \{1,\cdots,d\}$. To check that $\Lambda_{3,i}$ and $\tilde \Lambda_{3,i}$ agree on $C_0^\infty (E)$, it is enough to check that for $\varphi\in C_0^\infty (E)$, $\Lambda_{3,i}(\varphi)(t,x)=\tilde\Lambda_{3,i}(\varphi)(t,x)$ $dtdx$-a.e. Let $\varphi\in C_0^\infty (E)$. For $k\in \N$, we define, \[ \varphi_k(t,x) =\sum_{\val{n}\leq k} He_n(x/\sqrt{t}) (n!)^{-1} \int_{\R^d} dy\; p(t,y)He_n(y/\sqrt{t})\varphi(t,y). \] The sequence $(\varphi_k,k\geq 0)$ converges in $L^2(p)$ to $\varphi$. If $x\in \R^d$, $y\in \R$, $i \in \{1,\cdots,d\}$, we denote by $z=\hat x^i_y$ the element of $\R^d$ such that $z_i=y$ and $z_j=x_j$ for $j\neq i$. Since $\Lambda_{3,i}(f)(t,x)=-t^{-1} x_i \partial_i \Lambda(f)(t,x)$ for $f\in \ca\cup C^\infty _0(E)$, we see that an integration by parts gives \begin{equation} \label{eq:integbp} \int_0^{x_i} dy\; \Lambda_{3,i}(f)(t,\hat x^i_y)= -t^{-1} x_i \Lambda(f)(t,x)+\int_0^{x_i} dy\; t^{-1} \Lambda (f)(t,\hat x^i_y). \end{equation} For short we write $P_i(f)$ for the operator $P_i(f)(t,x)=\int_0^{x_i} dy\; f(t,\hat x^i_y)$. Let $R>0$ and $T>\varepsilon>0$ be fixed. Let $Q=[\varepsilon,T]\times [-R,R]^d$. The heat kernel $p$ is bounded below and above on $Q$ by positive constant, say $c_Q$ and $C_Q$. Using Cauchy-Schwarz inequality we have \[ \norm{\ind_Q P_i(f)}_{(p)}^2 \leq C_Q R^2 \iint_Q dtdx \; f(t,x)^2 \leq C_Q c_Q^{-1} R^2 \norm{f}_{(p)}^2. \] Thus the operator $\ind_Q P_i$ is continuous from $L^2(p)$ to $L^2(p)$. Thanks to Lemma \ref{lem:opL} and the above first step, we get that the sequences $(\ind_Q P_i(\Lambda_{0}(\varphi_k)),k\geq 0)$ and $(\ind_Q P_i(\Lambda_{3,i}(\varphi_k)),k\geq 0)$ converge in $L^2(p)$ respectively to $\ind_Q P_i(\Lambda_{0}(\varphi))$ and $\ind_Q P_i(\tilde\Lambda_{3,i}(\varphi))$. Notice also that $(\ind_Q \Lambda (\varphi_{\sigma(k)}), k\geq 0)$ converges in $L^2(p)$ to $\ind_Q \Lambda (\varphi)$. Thus, there is a subsequence $(\sigma(k),k\geq 0)$ such that the sequences $(\ind_Q P_i(\Lambda_{0}(\varphi_{\sigma(k)})),k\geq 0)$, $(\ind_Q P_i(\Lambda_{3,i}(\varphi_{\sigma(k)})),k\geq 0)$ and $\left(\Lambda (\varphi_{\sigma(k)}),k\geq 0\right)$ converge $dtdx$-a.e. respectively to $\ind_Q P_i(\Lambda_{0}(\varphi))$, $\ind_Q P_i(\tilde\Lambda_{3,i}(\varphi))$ and $\Lambda (\varphi)$. Now \reff{eq:integbp} holds for $f=\varphi_{\sigma(k)}$, this means that for $(t,x)\in Q$, \[ P_i (\Lambda_{3,i}(\varphi_{\sigma(k)}))(t,x)=-t^{-1} x_i \Lambda(\varphi_{\sigma(k)})(t,x)+P_i (\Lambda_{0}(\varphi_{\sigma(k)}))(t,x). \] Taking the limit we get that $dtdx$-a.e. in $Q$, \[ P_i (\tilde\Lambda_{3,i}(\varphi))(t,x)=-t^{-1} x_i \Lambda(\varphi)(t,x)+P_i (\Lambda_{0}(\varphi))(t,x). \] Since $R,T,\varepsilon$ are arbitrary, the above equality holds $dtdx$-a.e. in $E$. Since \reff{eq:integbp} holds also for $f=\varphi$, we deduce that $dtdx$-a.e., \[ \int_0^{x_i} dy \; \Lambda_{3,i}(\varphi)(t, \hat x^i_y)=\int_0^{x_i} dy \; \tilde \Lambda_{3,i}(\varphi)(t, \hat x^i_y). \] Hence we have $dtdx$-a.e., $\Lambda_{3,i}(\varphi)(t,x)=\tilde \Lambda_{3,i}(\varphi)(t,x)$. 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