\documentstyle[11pt,twoside]{article} \topmargin=-10mm \oddsidemargin=0mm \evensidemargin=0mm \textheight=230mm \textwidth=130mm \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{prop}[theorem]{Proposition} \newtheorem{corol}[theorem]{Corollary} \newtheorem{obs}[theorem]{Remark} \newcommand{\pn}{\par\noindent} \newcommand{\med}{\medskip} \renewcommand{\c}{\centerline} \newcommand{\bs}{\backslash} \newcommand{\til}{\tilde} \newcommand{\ov}{\overline} \newcommand{\wtil}{\widetilde} \newcommand{\what}{\widehat} \newcommand{\rta}{\rightarrow} \newcommand{\Lrta}{\Longleftrightarrow} \newcommand{\ve}{\varepsilon} \renewcommand{\a}{\alpha} \renewcommand{\b}{\beta} \newcommand{\g}{\gamma} \renewcommand{\d}{\delta} \newcommand{\ee}{{\rm e}} \newcommand{\s}{\sigma} \newcommand{\R}{{\bf R}} \renewcommand{\ov}{\overline} \newcommand{\bl}{{\hbox{\vrule height 1.5mm width 2mm}}\rm} %\newcommand{\hs#1}{\hskip -#1pt} \newcommand{\qq}{\qquad} \newcommand{\q}{\quad} \newcommand{\Om}{\Omega} \newcommand{\D}{\Delta} \newcommand{\Di}{ {\cal D}} \documentstyle[12pt]{article} \hfuzz=1000pt \newtheorem{Theorem}{Theorem}[section] \newtheorem{Definition}[Theorem]{Definition} \newtheorem{Proposition}[Theorem]{Proposition} \newtheorem{Lemma}[Theorem]{Lemma} \newtheorem{Corollary}[Theorem]{Corollary} \newtheorem{Remark}[Theorem]{Remark} \newtheorem{Example}[Theorem]{Example} \newtheorem{Hypothesis}[Theorem]{Hypothesis} \makeatletter \newif\ifmsbmloaded@ \def\loadmsbm{\msbmloaded@true \font\tenmsb=msbm10 scaled 1\@ptsize00 \font\sevenmsb=msbm7 scaled 1\@ptsize00 \font\fivemsb=msbm5 scaled 1\@ptsize00 \alloc@8\fam\chardef\sixt@@n\msbfam \textfont\msbfam=\tenmsb \scriptfont\msbfam=\sevenmsb \scriptscriptfont\msbfam=\fivemsb } \def\nonmatherr@#1{\errmessage% {LateX error: \string#1\space allowed only in math mode}} \def\Bbb{\relax\ifmmode\expandafter\Bbb@\else \expandafter\nonmatherr@\expandafter\Bbb\fi} \def\Bbb@#1{{\Bbb@@{#1}}} \def\Bbb@@#1{\fam\msbfam\relax#1} \@addtoreset{equation}{section} \def\theequation{\thesection.\arabic{equation}} \makeatother \loadmsbm \def\R{\Bbb R} \def\N{\Bbb N} \def\Z{\Bbb Z} \def\C{\Bbb C} \def\E{\Bbb E} \def\P{\Bbb P} \def\H{\Bbb H} \def\ds{\displaystyle} \textwidth=5in \textheight=8in \begin{document} \def\qed{\hbox{\hskip 6pt\vrule width6pt height7pt depth1pt \hskip1pt}\bigskip} \begin{center} {\Large \bf Maximal $L^p$ regularity for elliptic equations with unbounded coefficients} \footnote{to appear in ``Nonlinear Analysis, Theory, Methods \& Applications} \vspace{10mm} Giuseppe Da Prato, Scuola Normale Superiore di Pisa Piazza dei Cavalieri 7, 56126, Pisa, Italy Fax: 39-050-563513, daprato@sns.it\vspace{3mm} \noindent Vincenzo Vespri, Dipartimento di Matematica Ulisse Dini Viale Morgagni 67A, 50134, Firenze, Italy Fax: 39-055-4222695, vespri@dma.unifi.it \end{center}\vspace{4mm} \footnotesize {\bf Abstract}: We consider an elliptic equation on $R^n$ of the form $\lambda \varphi-\frac{1}{2}\;\Delta \varphi +\langle DU,D\varphi\rangle=f$ with the potential $U$ regular but unbounded. We prove a maximal regularity result in the space $L^p(R^n, \nu)$ where $\nu$ is an invariant measure. {\bf Key words}: elliptic equations, unbounded coefficients, maximal regularity, invariant measures. \normalsize \section{Introduction and setting of the problem} We are concerned with the following equation in $\R^{n}:$ \begin{equation} \label{e1.1} \lambda \varphi -N_0\varphi = f,\;\; \lambda>0, \end{equation} where $N_0$ is the elliptic operator defined by $$ N_0\varphi=\frac{1}{2}\;\Delta \varphi -\langle DU,D\varphi\rangle,\; \varphi \in C^{\infty}_{0}(\R^{n}), $$ and $U:\R^{n}\to \R$ is a function satisfying the following assumption. \begin{Hypothesis} \label{h1.1} \begin{enumerate} \item[(i)] $U$ is of class $C^3$. \item[(ii)] $\ds{\int_{\R^{n}}^{}e^{-2U(y)}dy<+\infty}$. \end{enumerate} \end{Hypothesis} It is important to notice that $N_0$ is a symmetric operator in the space $L^{2}(\R^n,\nu),$ where $\nu$ is the Borel probability measure on $\R^{n}$ defined as $$\nu(dx)=\frac{e^{-2U(x)}dx}{\int_{\R^n}^{}e^{-2U(y)}dy}. $$ Indeed the following identity can be easily checked: \begin{equation} \label{e1.2} \int_{\R^n}N_0\varphi \psi d\nu =-\frac{1}{2}\;\int_{\R^n} \langle D\varphi ,D\psi \rangle d\nu, \end{equation} for all $\varphi ,\psi \in C^\infty_0(\R^{n}).$ See Lemma \ref{l1.2} below. The operator $N_{0}$ naturally arises in studying a dynamical system described by a differential equation of potential type perturbed by a white noise, $$dX=-DU(X)dt+dW(t),\; X(0)=x, $$ where $W(t)$ is a standard Brownian motion on $\R^n.$ In fact, let us consider the transition semigroup defined by $$P_t\varphi (x)=\E[\varphi (X(t,x))],\; \varphi \in C_{b}(\R^{n}), $$ where $C_{b}(\R^{n})$ is the space of all unformly continuous and bounded functions on $\R^{n},$ $X(t,x)$ is the solution of the stochastic equation above and $\E$ denotes the expectation. Then, in a sense to be precised, $N_{0}$ is the infinitesimal generator of $P_t,$ and $\nu$ is an invariant measure for $P_t:$ $$ \int_{\R^n}P_t\varphi d\nu=\int_{\R^n}\varphi d\nu,\;\varphi \in C_{b}(\R^{n}). $$ If $DU$ is continuous and bounded then wellposedness for the elliptic equation (\ref{e1.1}), or for the corresponding parabolic equation, is well known, see \cite{LUS}. However the boundedness assumption for $DU$ is too restrictive for the applications. A general existence result for equation (\ref{e1.1}) with $DU$ was proved by S. Ito, see \cite{Ito}; however uniqueness of the solutions to (\ref{e1.1}) is not trivial, see e.g. \cite{Krylov}. Some cases of wellposedness (for an equation similar to (\ref{e1.1}) but including an additional term $V\varphi$) in suitable weighted spaces, are presented in \cite{Besala}, \cite{CV}, \cite{CV1}. In the past few years many papers have been devoted to study equation (\ref{e1.1}), even for non regular potentials $U$ and also in infinite dimensional spaces $H.$ We recall the approach through the Dirichlet forms theory, see \cite{AH}, \cite{AKR}, \cite{Ma} and the references therein, and through the parabolic criterion of Berezanskii, \cite{Bere}. Another approach was introduced in \cite{Bumi}, where essential \break self--adjointness of $N_0$ was proved by showing the density of the range of $\lambda I -N_0 $ for any $\lambda>0.$ From Hypothesis \ref{h1.1} it follows that $N_0$ is essentially self--adjoint on $L^{2}(\R^n,\nu)$, see e.g. \cite{DPG}. We shall denote by $N_{2}$ its closure on $L^{2}(\R^n,\nu)$. Moreover from (\ref{e1.2}) it follows that the domain of the square root of $-N_{2},$ is $W^{1,2}(\R^n,\nu).$ A more difficult task is to characterize the domain of $N_{2}.$ At our knowledge such a characterization is known in two cases: when $DU$ is linear (see \cite{L},\cite{DP1}) and when $U$ is convex (see \cite{DP2}, \cite{DPG}). In both cases the domain of $N_2$ is given by $$ D(N_2)=W^{2,2}(\R^n,\nu). $$ Let now $p>1$. Then $N_0$ is dissipative in $L^p(\R^n,\nu)$, see Lemma \ref{l1.2} below, and so it is closable. Let us denote by $N_p$ its closure. The main aim of this paper is to prove that $N_p$ is $m$--dissipative and that its domain is given by \begin{equation} \label{e1.3} D(N_p)=W^{2,p}(\R^n,\nu). \end{equation} This result seems to be unknown, unless $U$ is a quadratic form; see the paper \cite{CG}, where (\ref{e1.3}) was proved (even in an infinite dimensional Hilbert space $H$) by using a generalization of the Littlewood--Paley--Stein Theory. We are able to prove (\ref{e1.3}) under quite strong assumptions on $U.$ Rougly speaking we can cover the case when $U$ and its derivatives up to the second order have polynomial growth. Let us explain our method. Consider equation (\ref{e1.1}) where $\lambda >0,$ $f\in L^{p}(\R^n,\nu)$ and $p>1.$ Setting $$v(x)=e^{-\frac{2}{p}U(x)}\varphi(x),\;g(x)=e^{-\frac{2}{p}U(x)}f(x),\;\;x\in \R^n, $$ equation (\ref{e1.1}) reduces to % % equation % \begin{equation} \label{e1.4} \lambda v-\frac{1}{2}\;\Delta v+\frac{p-2}{p}\; \langle Dv,DU \rangle+G_pv=g, \end{equation} where $$G_p= \frac{2(p-1)}{p^{2}}\;|DU|^{2}-\frac{1}{p}\;\Delta U. $$ We shall prove, under suitable assumptions, existence and uniqueness of a solution of equation (\ref{e1.4}) together with maximal regularity, using a result due to P. Cannarsa and V. Vespri, see \cite{CV}. Then, going back to equation (\ref{e1.1}), we are able to show that $N_p$ is $m$--dissipative and that (\ref{e1.3}) holds. We end this section by proving the above mentionned lemma and recalling the result of \cite{CV}. \begin{Lemma} \label{l1.2} Assume that Hypothesis \ref{h1.1} holds. Then for any $\varphi ,\psi \in C^{\infty }_{0}(\R^n)$ the following identity holds. % % equation % \begin{equation} \label{e1.5} \int_{\R^n}^{}(N_{0}\varphi )\psi d\nu=-\frac{1}{2}\;\int_{\R^n}^{}\langle D\varphi ,D\psi \rangle d\nu. \end{equation} Moreover, for $p>2$, and for any $\varphi\in C^\infty_0(\R^n)$, \begin{equation} \label{e1.6} \int_{\R^n}^{}(N_{0}\varphi )|\varphi |^{p-2}\varphi d\nu=-\frac{1}{2}\;\int_{\R^n}^{} |D\varphi |^{2} |\varphi |^{p-2}d\nu. \end{equation} \end{Lemma} \noindent {\bf Proof}---Set % % displaymath % \begin{displaymath} \rho (x)=\frac{e^{-2U(x)}}{\int_{\R^n}^{}e^{-2U(y)}dy}, \; x\in H. \end{displaymath} Then we have % % displaymath % \begin{displaymath} \begin{array}{l} \ds{\int_{\R^n}^{} (N_0\varphi) \psi d\nu=\frac{1}{2}\;\int_{\R^n}^{}\Delta \varphi (\psi \rho )dx-\int_{\R^n}^{}\langle DU,D\varphi \rangle \psi d\nu }\\ \\ \ds{=-\frac{1}{2}\;\int_{\R^n}^{}\langle D\varphi ,D\psi \rangle d\nu-\frac{1}{2}\;\int_{\R^n}^{}\langle D\varphi , D\log \rho \rangle \psi d\nu-\int_{\R^n}^{}\langle DU,D\varphi \rangle \psi d\nu }\\ \\ \ds{= -\frac{1}{2}\;\int_{\R^n}^{}\langle D\varphi ,D\psi \rangle d\nu. } \end{array} \end{displaymath} Thus (\ref{e1.5}) is proved. Now (\ref{e1.6}) follows setting $\psi=|\varphi |^{p-2}\varphi $ in (\ref{e1.5}) .\qed \begin{Remark} {\em From (\ref{e1.5}) it follows immediately that $N_{0}$ is symmetric on $L^{2}(\R^n,\nu).$ Moreover, using the duality mapping on $L^{p}(\R^n,\nu)$ and formula (\ref{e1.6}), we can show easily that $N_p$ is dissipative on $L^{p}(\R^n,\nu)$ for $p\ge 2.$ Moreover, by an argument due to A. Eberle, see \cite{Eberle}, we see that $N_p$ is also dissipative on $L^{p}(\R^n,\nu)$ for $p\in [1,2].$ } \end{Remark} To state the result of \cite{CV} we need the following assumption. \begin{Hypothesis} \label{h1.3} \begin{enumerate} \item[] \item[(i)] There exists $k>0$ such that $G_{p}(x)+k\ge 1,\; x\in \R^n,$ and \begin{displaymath} |DG_{p}(x)|=o(|G_{p}(x)+k|^{3/2}), \; x\in \R^n, \end{displaymath} that is for any $\varepsilon >0$ there exists $k_{\varepsilon }>0$ such that % % displaymath % \begin{displaymath} |DG_{p}(x)|\le \varepsilon (|G_{p}(x)+k|^{3/2})+k_{\varepsilon },\; x\in \R^n. \end{displaymath} \item[(ii)] There exists $b\in [0,2)$ such that % % displaymath % \begin{displaymath} |DU(x)|^{2}\le b^{2}(G_{p}(x)+k),\; x\in \R^n. \end{displaymath} \end{enumerate} \end{Hypothesis} The following result was proved in \cite{CV}. \begin{Theorem} \label{t1.4} Assume that $DU$ is continuous and that Hypothesis \ref{h1.3} holds. Then for all $\lambda >k$ there exists a unique solution $v\in W^{2,2}(\R^n,dx)$ of (\ref{e1.3}). Moreover % % equation % \begin{equation} \label{e1.7} D_{i}v\;\sqrt{G_{p}(x)+k},\; \mbox{\rm and}\;D_{i}D_{j}(G_{p}(x)+k)\in L^{p}(\R^n,dx), \end{equation} for all $i,j=1,...,n.$ \end{Theorem} \section{Results} We shall assume, besides Hypotheses \ref{h1.1} and \ref{h1.3}, that \begin{Hypothesis} \label{h2.1} There exists $k_{1}>0$ such that \begin{displaymath} |D_{i}D_jU(x)|\le k_{1}(G_{p}(x)+k),\;\; x\in H,\;i,j=1,...,n. \end{displaymath} \end{Hypothesis} Roughly speaking Hypothesis \ref{h2.1} means that $U, D_{i}U, D_{i,j}U,$ $i,j=1,...,n,$ behave like a polynomial, see Examples \ref{Ex2.3} and \ref{Ex2.7} below. We can now prove the main result of the paper. \begin{Theorem} \label{t2.2} Assume that Hypotheses \ref{h1.1}, \ref{h1.3} and \ref{h2.1} hold. Then $N_{p}$ is $m$--dissipative in $L^{p}(\R^n,\nu)$ and we have $D(N_{p})=W^{2,p}(\R^n,\nu).$ \end{Theorem} {\bf Proof}---We are clearly in position to apply Theorem \ref{t1.4}. Thus if $\lambda >k,$ there is a unique solution $v\in W^{2,2}(\R^n,dx)$ of (\ref{e1.3}), and (\ref{e1.7}) holds. Moreover, using Hypothesis \ref{h2.1} and the obvious identities $$D_{i}\varphi =\left[D_{i}v+\frac{2}{p}\;(D_{i}U)v \right]e^{\frac{2}{p}U},\;i=1,...,n, $$ and $$ D_i\varphi D_j\varphi =\left[D_{i}D_{j}v+\frac{2}{p}\;(D_{i}UD_{j}v+D_{j}UD_{i}v)+\frac{4}{p^{2}} \;D_{i}UD_{j}U v \right]e^{\frac{2}{p}U}, $$ we see that $\varphi (x)=e^{\frac{2}{p}U(x)}v(x)$ is a solution of (\ref{e1.1}) and belongs to $W^{2,p}(\R^n,\nu).$ Moreover, $\varphi $ belongs to the domain of the closure $N_{p}$ of $N_{0}$ on $L^{p}(\R^n,\nu).$ In fact from the proof of \cite{CV} it follows that $C^{\infty }_{0}(\R^n)$ is a core for the linear operator $$ Lv=\frac{1}{2}\;\Delta v+\frac{p-2}{p}\; \langle Dv,DU \rangle+G_pv $$ on $L^p(\R^n,dx)$. Since the transformation $$u(x)=e^{\frac{2}{p}U(x)}\varphi(x), \;x\in \R^n, $$ maps $C^{\infty }_{0}(\R^n)$ into itself, it follows that $C^{\infty }_{0}(\R^n)$ is also a core for $N_{p}.$ To end the proof it remains to remove the condition $\lambda >k.$ This can be easily done because $N_{0}$ is dissipative on $L^{p}(\R^n,\nu)$ by Lemma \ref{l1.2}. The proof is complete.\qed \begin{Remark} \label{r2.2bis} {\em Assume that $p=2,$ and that Hypotheses \ref{h1.1}, \ref{h1.3}--(i)--(ii) and \ref{h2.1} hold. Then Hypothesis \ref{h1.3}--(iii) also holds. Under these assumptions $U$ is not necessarily convex. Thus the conclusion of the theorem seems to be new. } \end{Remark} Let us give now an example. \begin{Example} \label{Ex2.3} {\em Let $n=1$ and % % displaymath % \begin{displaymath} U(x)=\frac{1}{4}\;x^{4},\;\; x\in H. \end{displaymath} Then equations (\ref{e1.1}) and (\ref{e1.4}) read as follows $$ \lambda \varphi -\frac{1}{2}\;\varphi _{xx}+\frac{1}{2}\;x^{3}\varphi_{x}=f, $$ and $$ \lambda v-\frac{1}{2}\;v_{xx}+\frac{p-2}{p}\;x^{3}v_{x}+\frac{1}{p^{2}}\;\left( 2(p-1)x^{6}-3px^{2} \right)=g, $$ respectively. Thus it is easy to check that Hypotheses \ref{h1.1}, \ref{h1.3}--(i)--(ii), and \ref{h2.1} are fulfilled. As for as Hypothesis \ref{h1.3}--(iii) is concerned, it holds provided \begin{displaymath} 4-\sqrt{8}0$ such that if $\lambda >k$ we have % % displaymath % \begin{displaymath} \int_{\R^n}^{}(G_{p}+k)^{p}|v|^{p}dx\le c. \end{displaymath} Then for any $M>0$ we have obviously \begin{equation} \label{e2.6} (\lambda-k) v-\frac{1}{2}\;\Delta v+\frac{p-2}{p}\; \langle Dv,DU \rangle+M(G_p+k)v=g+(M-1)G_{p}v. \end{equation} In view of Lemma \ref{l2.4} we have $g+(M-1)G_{p}v\in L^{p}(\R^n,dx).$ Moreover, given $b\in [0,2),$ by an appropriate choice of $M,$ we get % % displaymath % \begin{displaymath} |DU(x)|^{2}\le b^{2}M(G_p(x)+k),\; x\in H. \end{displaymath} Now we can apply Theorem \ref{t1.4} and conclude that for any $\lambda >k$ equation (\ref{e1.4}) has a unique solution $v\in W^{2,p}(\R^n,dx),$ such that (\ref{e1.7}) holds. Consequently equation (\ref{e1.1}) has a unique solution in $W^{2,p}(H,\mu ).$ Finally the restriction $\lambda >k$ can be removed as before since $N_{p}$ is dissipative in $L^{p}(\R^n,\nu).$\qed \begin{Example} \label{Ex2.7} {\em Let $H=\R^{n},\;n, m\in \N,$ and let % % displaymath % \begin{displaymath} U(x)=\frac{1}{4}\;|x|^{2m}. \end{displaymath} Then it is easy to see that all hypotheses of Theorem \ref{t2.6} are fulfilled. } \end{Example} \begin{Remark} \label{r2.9} {\em Assume that the assumptions of Theorem \ref{t2.2} or \ref{t2.6} hold. Then $P_{t}=e^{tN_{2}}$ is a Markov semigroup on $L^2(\R^n,\nu).$ By \cite[Theorem 1.4.2]{Davies} it follows that $e^{tN_{p}}$ is an analytic semigroup in $L^p(\R^n,\nu)$ for any $p>1.$ } \end{Remark} \begin{Remark} \label{r2.9} {\em Let us consider equation (\ref{e1.1}) where $N_0$ is the elliptic operator $$ N_0\varphi=\frac{1}{2}\; \mbox{\rm div}\; [AD\varphi] -\langle A^{-1}DU,D\varphi\rangle\; \varphi \in C^{\infty}_{0}(\R^n), $$ where $U:\R^{n}\to \R$ is a function fulfilling Hypothesis \ref{h1.1} and $A\in C(\R^{n};L(\R^{n}))$ is such that % % displaymath % \begin{displaymath} \langle A\xi ,\xi \rangle\ge \alpha |\xi |^{2},\;\; \xi \in \R^{n}, \end{displaymath} for some $ \alpha >0.$ Then it is easy to see that $N_0$ is again a symmetric operator in the space $L^{2}(\R^n,\nu),$ where in this case $\nu$ is the Borel probability measure on $H$ defined as $$\nu(dx)=\frac{e^{-2U(x)}dx}{\int_{\R^n}^{}e^{-2U(y)}dy}, $$ and that all previous considerations can be adapted to cover this more general situation. } \end{Remark} \begin{Remark} \label{r2.10} {\em Let us consider equation (\ref{e1.1}), where $N_0$ is the elliptic operator $$ N_0\varphi=\frac{1}{2}\; \mbox{\rm div}\; [AD\varphi] -\langle B(x),D\varphi\rangle,\; \varphi \in C^{\infty}_{0}(\R^n), $$ $B $ is a regular vector field, and $A $ is as in the previous remark. Then the existence of an invariant measure $\nu,$ absolutely continuous with respect to the Lebesgue measure and such that \begin{equation} \label{e2.11} \int_HN_0\varphi \varphi d\nu =-\frac{1}{2}\;\int_H\langle AD\varphi, D\varphi \rangle d\nu, \end{equation} for all $\varphi \in C^\infty_0(\R^n),$ is well known, see e.g. \cite{DPG}. However in this case we do not know explicity the density $\rho =d\nu/dx.$ Moreover only local regularity results for $\rho $ are available, see \cite{BKR}. So it seems difficult to generalize the previous results to cover this case. } \end{Remark} \section{ACKNOWLEDGEMENT} The authors are indebted to Alessandra Lunardi for useful comments on the paper. \footnotesize \begin{thebibliography}{9} \bibitem{AH} ALBEVERIO S. \& HOEGH--KROHN (1977) {\it Dirichlet forms and diffusion processes on rigged Hilbert spaces,} Z. Wahrsch. verw. Geb. {\bf 40}, 1--57. \bibitem{AKR} ALBEVERIO S., KONDRATIEV V. \& R\"OCKNER (1995) {\it Dirichlet operators via stochastic analysis,} J. Funct. Analysis, {\bf 128}, n.1, 102--138. \bibitem{Bere} BEREZANSKII Yu. M. (1986) {\sc Self--adjoint operators in spaces of functions of infinitely many variables,} Transl. Math. Monographs, Vol. 63, Amer. Mat. Soc., Providence RI. \bibitem{Besala} BESALA P. (1975) {\it On the existence of a fundamental solution for a parabolic equation with unbounded coefficients}, Ann. Polon. Math. {\bf 29}, 403-409. \bibitem{BKR} BOGACHEV V.I., KRYLOV N. V. \& R\"OCKNER (to appear){\it On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions.} \bibitem{CV} CANNARSA P. \& VESPRI V. (1985) {\it Generation of analytic semigroups by elliptic operators with unbounded coefficients}, SIAM J. Math. Anal., {\bf 18}, 3, 857--872. \bibitem{CV1} CANNARSA P. \& VESPRI V. (1988) {\rm Generation of analytic semigroups in the $L^p$ topology by elliptic operators in $\R^n$}, {\it Israel Journal of Mathematics}, {\bf 61}, 3, 235--255. \bibitem{CG} CHOJNOWSKA--MICHALIK \& GOLDYS B. (1999) {\it Symmetric Mehler semigroups in $L^p$: Littlewood--Paley--Stein inequalities and domains of generators,} S.N.S. Pisa, Preprint n. 15. \bibitem{DP1} DA PRATO G. (1997) {\it Regularity results for Kolmogorov equations on $L^{2}(H,\mu)$ spaces and applications,} Ukrainian Mathematical Journal, n.49, n. 3, 448--457. \bibitem{DP2} DA PRATO G. (1997) {\it Characterization of the domain of an elliptic operator of infinitely many variables in $L^{2}(\mu )$ spaces.} Rend. Mat. Acc. Lincei, s.9, v. 8, 101--105. \bibitem{Bumi} DA PRATO G. (1998), {\it The Ornstein--Uhlenbeck generator perturbed by the gradient of a potential,} Bollettino UMI, (8) I-B, 501--519. \bibitem{DPG} DA PRATO G. \& GOLDYS B. (to appear) {\it Elliptic operators on $\R^{d}$ with unbounded operators}, J. Differential Equations. \bibitem{Davies} DAVIES E. B. (1989) {\sc Heat kernels and Spectral Theory}, Cambridge University Press. \bibitem{Eberle} EBERLE A. (1999) {\it Uniqueness and non--uniqueness of singular diffusion operators,} Lecture Notes in Mathematics 1718, Berlin, Springer--Verlag. \bibitem{Ito} ITO S. (1957) {\it Fundamental solutions of parabolic differential equations and boundary value problems,} Jap. J. Math. {\bf 27}, 5-102. \bibitem{Krylov} KRYLOV N. V. (1991) {\sc Introduction to the theory of diffusion processes,} American Mathematical Society, vol.142. \bibitem{LUS} LADYZHENSKAJA O.A., SOLONNIKOV V.A. \& URAL'CEVA N.N.: (1968) {\sc Linear and quasi linear equations of parabolic type,} Transl. Math. Monographs, AMS, Providence. \bibitem{L} LUNARDI A. (1997) {\it On the Ornstein-Uhlenbeck operator in $L^2$ spaces with respect to invariant measures}, Trans. Amer. Math. Soc. {\bf 349}, 155--169. \bibitem{Ma} MA Z. M. \& R\"OCKNER (1992) {\sc Introduction to the Theory of (Non Symmetric) Dirichlet Forms} , Springer--Verlag. \end{thebibliography} \end{document}