%This is an AMS-TeX file: Drichl10.tex %Drichlet problem for superdegenerate operators . %Note for Comptes Rendus. Corrected copy. % June 11, 1998, 9:57pm. %Authors: Denis Bell and Salah Mohammed % \input amstex \magnification=\magstep1 \documentstyle{amsppt} \catcode`\@=11 \def\logo@{} \catcode`\@=\active \baselineskip=18truept \NoBlackBoxes \TagsAsMath \TagsOnRight \parindent=.5truein \parskip=3pt \hsize=6.5truein \vsize=9.0truein \raggedbottom \loadmsam \loadbold \loadmsbm %\font\eightsmc=cmbx8 %Other math symbols \define \x{\times} \let \< = \langle \let \> = \rangle \let \| = \Vert %Greek letters \define \a {\alpha} \redefine \b {\beta} \redefine \dl {\delta} \redefine \D {\Cal D} \redefine \e {\epsilon} \define \F {\Cal F} %\define \G {\Gamma} \redefine \g {\gamma} \define \R {\bold R} \define \s {\sigma} \define \th {\theta} \redefine \O {\Omega} \redefine \o {\omega} \redefine \w {\omega} \redefine \t {\tau} \define \ub {\subset} \define \cl {\Cal L} \define \E {\Cal E} \define \A {\Cal A} \def \e {{\acuteaccent e}} \def \E {\graveaccent e} \def \k {\kern 5 pt} \def \n {\noindent} %\def \R {$\bold R^d$\kern 2 pt} %Other macros \redefine \i {\infty} \define \pt {\partial} \define \bl { L} \define \ube {\subseteq} \define\qd{{\unskip\nobreak\hfil\penalty50\hskip2em\vadjust{} \nobreak\hfil$\square$\parfillskip=0pt\finalhyphendemerits=0\par}} % \define \x{\times} \let \< = \langle \let \> = \rangle \let \| = \Vert %Greek letters \redefine \b {\beta} \redefine \dl {\delta} \redefine \D {\Cal D} \redefine \g {\gamma} \redefine \O {\Omega} \redefine \o {\omega} \redefine \w {\omega} %\redefine \R {\R} \redefine \t {\tau} %Other macros \redefine \i {\infty} %\define \pt {\partial} %\define \bl { L} %\define \ube {\subseteq} \font\oo=cmbx10 scaled \magstep1 \font \pp=cmb10 scaled \magstep1 % AmS-TeX macros for this file \def\hang{\hangindent\parindent} \def\ite{\par\hang\textindent} \def\iite{\par\indent \hangindent2\parindent \textindent} \font\Times = cmr10 \font\Timesb = cmbx10 \font\headfnt=cmbx8 %\font\it = Timesi \def \n {\noindent} \def \r {$\R^d$} \def \kk {\kern 10} \def \k {\kern 5} \def \x {$X_0, \dots, X_n\,$} \def \y {$X_1, \dots, X_n\,$} \def \h {H\"ormander} \def \O {\Omega} \topmatter \title The Dirichlet problem for superdegenerate differential operators \endtitle \author Denis R\. Bell and Salah E.-A. Mohammed \endauthor \affil Mathematical Sciences Research Institute\\ 1000 Centennial Drive\\ Berkeley\\ California 94720-5070. \medskip dbell\@unf.edu $\quad$ salah\@math.siu.edu \endaffil \abstract Let $L$ be an infinitely degenerate second-order linear operator defined on a bounded smooth Euclidean domain. Under weaker conditions than those of H\"ormander, we show that the Dirichlet problem associated with $L$ has a unique smooth classical solution. The proof uses the Malliavin calculus. At present, there appears to be no proof of this result using classical analytic techniques. \bigskip \centerline{\bf Le Probl\'eme De Dirichlet Pour } \centerline{ \bf Des Operateurs Differentiels Superd\'eg\'en\'er\'es} \noindent R\'E SUM\'E. Soit $L$ un op\e rateur lin\e aire d\e fini sur un domaine born\acuteaccent e r\e gulier de l'espace euclidien avec une d\e g\e n\e rescence infinie. Sous des conditions plus faibles que celles de H\"ormander, on montre que le probl\E me de Dirichlet {associ\e} \graveaccent a $L$ a une solution r\e guli\E re classique unique. La d\e monstration utilise le calcul de Malliavin. Il semble qu'il n'y ait \graveaccent a cette date aucune d\e monstration de ce r\e sultat par des techniques analytiques classiques. \endabstract \endtopmatter \def\rightheadline{\hfil The Dirichlet problem for superdegenerate operators \quad \folio} \def\leftheadline{\folio \quad D. R. Bell and S.-E. A. Mohammed \hfil} %\rightheadtext{The Dirichlet problem for superdegenerate operators} %\leftheadtext{D. R. Bell and S.-E. A. Mohammed} %\hrule %\headline={\centerline{\headfnt The Dirichlet problem for superdegenerate %operators\hfil}} %\headline={\centerline{\headfnt D. R. Bell and S.-E. A. Mohammed \hfil}} %\leftheadtext{} \bigskip \subheading{Version fran\c caise abr\'eg\'ee} \medskip \n Soit $D$ un domaine {born\e} r\e guli\E re de $\R^d$ dont la fronti\graveaccent ere $\partial D$ est r\e guli\E re. Soient \break $X_0, \dots, X_n$ des champs de vecteurs et $c$ une fonction \graveaccent a valeurs r\e elles, tous d\e finis et r\e guliers dans un voisinage ouvert de $\bar D$. Notons $L$ l'op\e rateur diff\e rentiel de second ordre $$L = \sum_{i = 1}^n X_i^2 + X_0 + c.$$ Dans un article ant\e rieur [4] (voir aussi [3]) les auteurs ont {donn\e} des conditions suffisantes pour que $L$ soit hypoelliptique qui autorisent la violation sur une hypersurface de $D$ des conditions de H\"ormander sur l'alg\E bre de Lie. Le type de d\e g\e n\e rescence {autoris\e} dans [4] est {caract\e ris\e} comme suit. \definition{D\e finition} Pour chaque $k \ge 0$, soit $X^{(k)}$ la matrice dont les colonnes sont \break $X_0, \dots, X_n$, et tous les champs de vecteurs obtenus \graveaccent a partir de $X_0, \dots, X_n$ en formant les crochets de Lie it\e r\e s jusqu' \graveaccent a l'ordre $k$. Soit $$ \lambda^{(k)} \hbox{ la plus petite valeur propre de} \kern 2 pt X^{(k)}X^{(k)t}$$ o\graveaccent u $^t$ d\e signe la transposition des matrices. On dit qu'une hypersurface $S \subset \R^d$ est sous-critique (relativement \graveaccent a $L$) au point $x \in S$ quand il existe un voisinage ouvert $U$ de $x$, un entier $k \ge 0$, et un $p \in (-1, 0)$ tels que $$ \lambda^{(k)}(y) \ge \exp\{-[\rho(y, S)]^p\}, \quad \forall y \in U, $$ o\graveaccent u $\rho(y, S)$ est la distance euclidienne de $y$ \graveaccent a l'hypersurface S. \enddefinition Soit $K$ l'ensemble des points de $D$ o\graveaccent u $L$ ne v\e rifie pas la condition de H\"ormander. Dans [4], les auteurs ont {montr\e} que $L$ est hypoelliptique \graveaccent a condition que $K$ soit contenu dans une hypersurface $S$ de $\R^d$ de classe $C^2$ qui est sous-critique et non-carat\e ristique en tout point de $K$. En particulier, ce th\e or\graveaccent eme donne {l'hypoellipticit\e} des op\e rateurs $$ {\partial^2\over\partial x^2} + \exp(-|x|^p){\partial^2\over\partial y^2} + {\partial^2\over\partial z^2}, \quad p \in (-1, 0) $$ qui ont {\e t\e} etudi\e s par Kusuoka et Stroock dans [8]. Des op\e rateurs diff\e rentiels qui ont des d\e g\e n\e r\e scences d'ordre infini seront appel\e s {\it superd\e g\e n\e r\e s}. Le but de cet article est d'annoncer des r\e ultats qui donnent l'existence d'une solution r\e guli\E re au probl\E me de Dirichlet pour des op\e rateurs superd\e g\e n\e r\e s. Les r\e sultats sont les suivants. \proclaim {Th\e or\E me 1} Supposons que l'ensemble $K \cup \partial D$ est contenu dans une hypersurface $S$ de classe $C^2$ qui est sous-critique et non-charact\e ristique en chaque point de $K \cup \partial D$. Soient $f \in C^\infty(\bar D)$ et $g \in C^\infty(\partial D)$ et soit $u$ une solution faible (au sense des distributions) du probl\E me de Dirichlet sur $D$ $${{Lu = f \quad dans \quad D} \atop {\gamma_0 u = g \quad sur \quad \partial D}} \Biggr \} \eqno (1) $$ o\graveaccent u $\gamma_0$ repr\e sente la trace sectionnelle sur $\partial D$. Alors $u \in C^\infty (\bar D)$. \endproclaim \proclaim {Th\e or\E me 2} Supposons que les hypoth\E ses du Th\e or\E me 1 sont satisfaites et que de plus \hfil\break (a) $c \le 0$ sur $\bar D$ \hfil\break (b) Il existe $1 \le k \le d$ et $a > 0$ tels que $\sum_{i = 1}^n ^2 \ge a$ pour tout $x \in \bar D$, o\graveaccent u $e_k$ est le $k^{i\grave e me}$ vecteur de la base standard de $\R^d$. Alors le probl\E me de Dirichlet (1) admet une solution $C^\infty$ unique $u$ sur $\bar D$. \endproclaim Le Probl\E me de Dirichlet pour des op\e rateurs d\e g\e n\e r\e s $L$ a {\e t\e} {\e tudi\e} par divers auteurs (voir Bony [1], Derridj [6], Jerison [7], et Cattiaux [5]) sous l'hypoth\E se que $L$ satisfasse \graveaccent a la condition de H\"ormander en tout point. A la connaissance des auteurs, les r\e sultats ci-dessus sont les premiers \graveaccent a \e tablir l'existence d'une solution r\e guli\E re au probl\E me de Dirichlet sous des hypoth\E ses qui permettent des d\e g\e n\e r\e scences d'ordre infini. En particulier, il semble que ces r\e sultats ne peuvent \^etre obtenus \graveaccent a partir des techniques classiques. \medskip \demo{ Sch\e ma de d\e monstration de th\e or\E me 2} On consid\E re le processus de diffusion $d$-dimensionel {donn\e} par l'\e quation diff\e rentielle stochastique de Stratonovich $$ \left. \aligned d\xi^x (t) &= X_0 (\xi^x (t)) \, dt + \sum _{i = 1}^n X_i (\xi^x (t))\circ d W_i(t)\\ \xi^x(0)&=x \in D \endaligned \right \}\eqno(2) $$ {associ\e} \graveaccent a l'op\e rateur $L$. Alors sous les conditions de Th\e or\E me 2, une solution faible du probl\E me de Dirichlet (1) est donn\e e par $$ u(x) = E \biggl [ g(\xi^x( \tau (x))) \exp \biggl \{\int_0^{\tau(x)} c(\xi^x(s)\,ds \biggr \} \,dt\Big ] \kern 100 pt$$ $$\kern 130 pt -E \Big [ \int_0^{\tau(x)} f(\xi^x( t)) \exp \biggl \{\int_0^t c(\xi^x(s)\,ds \biggr \} \,dt \Big ] \eqno{(3)} $$ o\graveaccent u $\tau = \tau(x)$ est le premier temps de sortie de $D$ de la diffusion $\xi^x$. Sans perdre de {g\e n\e ralit\e} on peut supposer que $g \equiv 0$ sur $\partial D$, et pour simplifier prenons aussi $c \equiv 0$ ([5]). Donc il est suffisant de montrer que la fonction $$ u(x) = -E \Big [ \int_0^{\tau(x)} f(\xi^x( t))\,dt \Big ] \eqno (4) $$ est r\e guli\E re sur $\bar D$. Maintenant $L$ est hypoelliptique sur $D$ par [4] et $u$ est une solution faible de l'\e quation $Lu = 0$ dans $D$ ([10]). Pour prouver la {r\e gularit\e} de $u$ jusque $\partial D$, il faut d\e montrer des estimations uniformes sur les d\e riv\e es $D^ku(x), k \ge 1$, quand $x$ tend vers $\partial D$ en restant dans $D$. Ceci est fait combinant les techniques de [2], [5], et [4]. Le r\e sultat suivant joue un role essentiel \proclaim {Lemme} Supposons que les hyoth\E ses du th\e or\E me 1 sont satisfaites. Alors pour tout point $x_0 \in \partial D$, on peut trouver un voisinage $U$ de $x_0$ dans $\bar D$ et un diff\e omorphisme $F: \bar U \to F(\bar U) \subset [0,\infty) \times \R^{d - 1}$ ayant les propri\e t\e s suivantes: \item{(i)} $F(\bar U \cap \partial D) = F(\bar U) \cap (\{ 0\} \times \R^{d - 1})$ et $F(\bar U \cap D) = F(\bar U) \cap \R^{d + }$, o\graveaccent u $\R^{d + }:=$ \hfil\break $\{x:=(x_1,x_2,\cdots,x_d): x_1 > 0 \}$. \item{(ii)} Il existe $1 \le i \le n$ tels que $F^*(X_i) = (1, 0, \dots, 0)^t$, o\graveaccent u $$F^*(X_i)(x) := DF(F^{-1}(x))X_i(F^{-1}(x)).$$ \item{(iii)} Les champs de vecteurs $F^*(X_j), j \ne i, 0 \le j \le n$, sont de la forme $(0, Y)^t$ o\graveaccent u $Y$ est une fonction r\e guli\E re de $\R^d$ dans $\R^{d - 1}$. \item{(iv)} L'ensemble $F(\bar U \cap \partial D)$ est sous-critique relativement \graveaccent a op\e rateur {transform\e} $$F^*(L) = \sum_{i = 1}^n F^*(X_i)^2 + F^*(X_0) + c \circ F^{-1}.$$ \endproclaim Le lemme ci-dessus permet de r\e duire le probl\E me au case {o\graveaccent u} $D$ est le demi-espace $\bold R^{d + }$. En utilisant la repr\e sentation stochastique (4), on applique ensuite les estimations de [4] au processus de diffusion $F(\xi)$ pour obtenir les estimations d\e sir\e es sur les d\e riv\e es $D^ku(x), k \ge 1$, pour $x$ proche de $\partial D$. L'id\e e est la suivante. Comme dans [2] et [5], on conditionne la diffusion par rapport \graveaccent a son temps de sortie de $\bold R^{d + }$. Ceci est n\e cessaire parce que le temps de sortie est p.p. une fonction irr\e guli\E re de $x$. Les estimations de $D^ku(x), k \ge 1$, dont on a besoin sont ensuite obtenues par int\e gration par parties partielle sur l'espace de Wiener {conditionn\e}, comme dans le calcul de Malliavin [9]. Les estimations obtenues par les auteurs dans [4] sont suffisament robustes pour \^etre applicables dans le contexte pr\e sent, et jouent un role crucial dans l'analyse qui suit. \qed \enddemo %\medskip Le th\e or\E me 1 se d\e montre comme ci-dessus. \medskip \remark{Remarque} Le referee a observ\acuteaccent e que l'on pouvait remplacer la condition b) dans le th\acuteaccent eor\graveaccent eme 2 par l'hypoth\graveaccent ese suivante: %\noindent \acuteaccent Pour tout $x \in \bar D$, il existe un chemin $\gamma: [0,\infty) \mapsto R^d$ tel que $\gamma(0) = x, \gamma$ quitte $\bar D$, et $\gamma' (t) - X_0(\gamma(t)) \in \hbox{Lie}(X_1, \dots, X_n)(\gamma(t))$ pour tout temps $t$ avant que $\gamma$ quitte $\bar D$. \endremark \bigskip %\vbox to .2truein{} Suppose that $D$ is a bounded regular domain in $\R^d\,$ with a smooth boundary $\pt D$. Assume that \x are vector fields and $c$ is a real-valued function, defined and smooth in an open neighborhood of $\bar D$. Let $L$ denote the second order differential operator % $$L = \sum_{i = 1}^n X_i^2 + X_0 + c.$$ % In an earlier article [4] (see also [3]) the authors gave a sufficient condition for the hypoellipticity of $L$ under hypotheses that allow H\"ormander's Lie algebra condition to fail on a hypersurface in $D$. The type of degeneracy allowed in [4] is characterized in terms of the following %\newpage \definition {Definition} For each $k \ge 0$, define $X^{(k)}$ to be a matrix with columns \x, and all vector fields obtained from \x by forming iterated Lie brackets up to order $k$. Define $$ \lambda ^{(k)} := \hbox{ smallest eigenvalue of } X^{(k)}X^{(k)t} $$ where $^t$ denotes matrix transpose. We say that a hypersurface $S \subset$ \r\kern 2 pt is {\it subcritical \/} (with respect to $L$) at $x \in S$ if there exists an open neighborhood $U$ of $x$, an integer $k \geq 0$, and $p \in (-1,0)$ such that $$ \lambda ^{(k)} (y) \geq \exp \{ - [\rho (y,S)]^p \}, \quad \forall y \in U, \eqno $$ where $\rho$ denotes the Euclidean distance between $y$ and the hypersurface $S$. \enddefinition \medskip Let $K$ denote the set of points in $D$ where $L$ fails to satisfy H\"ormander's condition. A ($C^1$) hypersurface $S$ is said to be {\it non-characteristic\/} (with respect to $L$) at $x \in K$ if at least one of the vector fields \y $\,$ is transversal to $S$ at $x$. In [4], the authors showed that $L$ is hypoelliptic provided $K$ is contained in a $C^2$ hypersurface $S$ of $\R^d$, such that $S$ is subcritical and non-characteristic at all points of $K$. In particular, this theorem asserts the hypoellipticity of the class of operators $$ {\partial^2\over\partial x^2} + \exp(-|x|^p) {\partial^2\over\partial y^2} + {\partial^2\over\partial z^2}, \kern 10 pt p \in (-1, 0) $$ that were studied by Kusuoka and Stroock in [8]. Differential operators that exhibit infinite-order degeneracy will be termed {\it superdegenerate\/}. In this article we announce results that ensure the existence of a smooth solution to the Dirichlet problem for a large class of superdegenerate operators. Our results are as follows %\newpage \proclaim{ Theorem 1} Suppose that the set $K \cup \partial D$ is contained in a $C^2$ hypersurface $S$ and that $S$ is subcritical and non-characteristic at all points of $K \cup \partial D$. Let $f \in C^\infty (\bar D)$, $g \in C^\infty(\partial D)$ and $u$ be a weak solution (in the sense of distributions) to the following Dirichlet problem on $\bar D$ % $${{Lu = f \quad in \quad D} \atop {\gamma_0 u = g \quad on \quad \partial D}} \Biggr \} \eqno (1) $$ % where $\gamma_0$ denotes the sectional trace of order 0 on $\partial D$. Then $u \in C^\infty(\bar D)$. \endproclaim \proclaim{Theorem 2} Suppose the hypotheses of Theorem 1 hold and in addition \hfil\break (a) $c \le 0$ on $\bar D$. \hfil\break (b) There exists $1 \le k \le d$ and $a > 0$ such that $\sum_{i = 1}^n ^2 \, \ge a, \,$for all $x \in \bar D$, where $e_k$ denotes the kth standard unit vector in \r.\hfil\break Then the Dirichlet problem (1) admits a unique smooth solution $u$ on $\bar D$. \endproclaim The Dirichlet problem for degenerate operators $L$ has been studied by several authors (cf. Bony [1], Derridj [6], Jerison [7], and Cattiaux [5]) under the assumption that $L$ satisfies \h's condition at all points. As far as the authors are aware, the above results are the first to establish the existence of a smooth solution to the Dirichlet problem under hypotheses that allow degeneracy of {\it infinite\/} order. In particular, it appears that these results are at present unavailable using classical techniques. \bigskip \demo{Outline of Proof of Theorem 2} Consider the $d$-dimensional diffusion process $\xi$ given by the Stratonovich stochastic differential equation % $$ \left. \aligned d\xi (t) &= X_0 (\xi (t)) \, dt + \sum _{i = 1}^n X_i (\xi (t))\circ d W_i(t)\\ \xi(0)&=x \in D \endaligned \right \}\eqno(2) $$ % associated with the operator $L$. Then by the conditions of Theorem 2, a weak solution of the Dirichlet problem (1) is given by % $$ %\align u(x) = E \biggl [ g(\xi^x( \tau (x))) \exp \biggl \{\int_0^{\tau(x)} c(\xi^x(s)\,ds \biggr \} \,dt\biggr ] -E \biggl [ \int_0^{\tau(x)} f(\xi^x( t)) \exp \biggl \{\int_0^t c(\xi^x(s)\,ds \biggr \} \,dt \biggr ] \tag{3} %\endalign $$ % where $\tau=\tau(x)$ is the first exit time of the diffusion $\xi^x$ from $D$. Without loss of generality, we may assume that $g \equiv 0$ on $\pt D$, and for simplicity we will also take $c \equiv 0$ ([5]). Therefore, it is sufficent to prove that the function % $$ u(x) = -E \Big [ \int_0^{\tau(x)} f(\xi^x( t))\,dt \Big ] \tag{4} $$ % is smooth on $\bar D$. Now $L$ is hypoelliptic on $D$ by [4] and $u$ is a weak solution of the equation $Lu=0$ in $D$ ([10]). Therefore $u$ is $C^\infty$ on $D$. In order to prove smoothness of $u$ up to $\partial D$, it is necessary to derive uniform bounds on the derivatives $D^{(k)} u(x), k \geq 1$, as $x$ approaches $\pt D$ through points of $D$. This is achieved by combining the techniques in [2], [5], and [4]. The following result plays a key role \proclaim {Lemma} Suppose the hypotheses of Theorem 1 hold. Then for each point $x_0 \in \partial D$, there exists a neighborhood $U$ of $x_0$ in $\bar D$ and a diffeomorphism $F: \bar U \to F(\bar U) \subset [0,\infty) \times \R^{d - 1}$ %$F: \bar U \mapsto \{x \in \R: x \ge 0\} \times \R^{d - 1}$ with the following properties \item{(i)} $F(\bar U \cap \partial D) = F(\bar U) \cap (\{ 0\} \times \R^{d - 1})$ and $F(\bar U \cap D) = F(\bar U) \cap \R^{d +}$, where $\R^{d + }:=\{x:=(x_1,x_2,\cdots,x_d): x_1 > 0 \}$. \item{(ii)} There exists $1 \le i \le n$ such that $F^*(X_i) = (1, 0, \dots, 0)^t$, where % $$F^*(X_i)(x):= DF(F^{-1}(x))X_i (F^{-1}(x)).$$ % \item{(iii)} The vector fields $F^*(X_j), j \ne i, 0 \le j \le n$, have the form $(0, Y)^t$ where $Y$ is a smooth function from $\R^d$ to $\R^{d - 1}.$ \item{(iv)} The set $F(\bar U \cap \partial D)$ is subcritical with respect to the transformed operator $$F^*(L) = \sum_{i = 1}^n F^*(X_i)^2 + F^*(X_0) + c \circ F^{-1}.$$ \endproclaim \n The above lemma reduces the problem to the case when $D$ is the half-space $\R^{d + }$. Using the stochastic representation (4), the estimates in [4] are then applied to the diffusion process $F(\xi)$ in order to obtain the desired estimates on $D^{(k)} u(x), k \geq 1$, at points close to $\partial D$. The idea is as follows. Following [2] and [5], we condition the diffusion $F(\xi)$ up to its exit time from $\R^{d + }$. This is necessary because the exit time is a.s. irregular as a function of $x$. The required estimates on $D^{(k)} u(x), k \geq 1$, are then obtained through a process of partial integration by parts on the conditioned Wiener space, in the manner of the Malliavin calculus [9]. The estimates obtained by the authors in [4] are sufficiently robust to be applied in the present context, and play a crucial role in the subsequent analysis. %\kern 5 pt \qed \enddemo Theorem 1 is proved by an argument similar to the above. \remark{Remark} The referee has observed that condition (b) in Theorem 2 may be replaced by the following hypothesis: For each $x \in \bar D$, there is a $C^1$ path $\gamma: [0,\infty) \to \R^d$ such that $\gamma(0)=x$, $\gamma$ leaves $\bar D$, and $\gamma'(t) -X_0(\gamma(t)) \in \hbox{Lie}(X_1, \cdots, X_n)(\gamma(t))$ for all times $t$ before $\gamma$ exits $\bar D$. \endremark \subheading{Acknowledgement} The authors are grateful to Guy David for the translation into French, and to the referee for helpful suggestions. The research of Denis Bell is supported in part by NSF grant DMS-9703852 and by MSRI, Berkeley, California. The research of Salah Mohammed is supported in part by NSF grants DMS-9503702, DMS-9703596, and by MSRI, Berkeley, California. \bigskip \bigskip \bigskip \centerline{\bf{ References}} \bigskip \baselineskip=14truept \parskip=3truept \medskip \item{[1]\quad} Bony J. M., Principe du maximum, in\'egalit\'e de Harnack et unicit\'e du probl\`eme de Cauchy pour les op\'erateurs elliptiques d\'eg\'en\'er\'es, {\it Ann. Inst. Fourier\/}, Grenoble, t. 19 (1969) 277-304. \medskip \item{[2]\quad} Ben-Arous G., Kusuoka S., Stroock D. W., The Poisson kernel for certain degenerate elliptic operators, {\it J. Funct. Anal.\/} 56, no. 2 (1984) 171--209. \medskip \item{[3]\quad} Bell D. R., Mohammed S.-E. A., Hypoelliptic parabolic operators with exponential degeneracies, {\it C. R. Acad. Sci. Paris\/}, t. 317, S\'erie I (1993) 1059-1064. \medskip \item{[4]\quad} Bell D. R., Mohammed S.-E. A., An extension of H\"ormander's theorem for infinitely degenerate second-order operators, {\it Duke Math J.\/}, 78, no. 3 (1995) 453-475. \medskip \item{[5]\quad} Cattiaux P., Calcul stochastique et op\'erateurs d\'eg\'ener\'es du second ordre II. Problem de Dirichlet, {\it Bull. Sc. math., 2$^e$ s\'erie\/} 115 (1991) 81-122. \medskip \item{[6]\quad} Derridj M., Un probl\`eme aux limites pour une classe d'op\'erateurs du second ordre hypoelliptiques {\it Ann. Inst. Fourier\/}, Grenoble, t. 21 (1971) 99-148. \medskip \medskip \item{[7] \quad} Jerison D., The Dirichlet problem for the Kohn Laplacian on the Heisenberg group I, {\it J. Funct. Anal.\/} 43, no. 1 (1981) 97-142. \medskip \item{[8] \quad} Kusuoka S., Stroock D. W., Applications of the Malliavin calculus, Part II, {\it Journal of Faculty of Science, University of Tokyo}, Sec. 1A, 32 (1985) 1-76. \medskip \item{[9] \quad} Malliavin P., Stochastic calculus of variations and hypoelliptic operators, {\it Proceedings of the International Conference on Stochastic Differential Equations, Kyoto}, Kinokuniya (1976) 195-263. \medskip \item{[10] \quad} Stroock D. W., Varadhan S. R. S., On degenerate elliptic-parabolic operators of second order and their associated diffusions, {\it Comm. Pure. Appl. Math.\/} 25 (1972) 651--713. \end \enddocument