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\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}<p<4+\sqrt{8}.
  \end{displaymath}
This example can be obviously generalized  taking for $U$  any
polynomial of even degree having positive leading coefficient. }
\end{Example}
As  shown by the example, the assumption \ref{h1.3}--(iii) is
quite restrictive. As we shall see, it can be removed   provided
an estimate of the $L^p(\R^n,\nu)$ norm of $G_pv$ be available.




    From now on
   we shall assume that $p\ge 2$ for simplicity. Let us start with a lemma.
   \begin{Lemma}
   \label{l2.4}
   Let $v$ be a solution of (\ref{e1.4}). Then the following identity holds:
   %
   % equation
   %
   \begin{equation}
   \label{e2.1}
   \begin{array}{l}
   \ds{(\lambda-k)

\int_{\R^n}^{}(G_{p}+k)^{p-1}|v|^{p}dx+\frac{p-1}{2}\;\int_{\R^n}^{}|Dv|^{2}|v|^{p-2}
(G_{p}+k)^{p-1}dx
   }\\
   \\
   \ds{+\int_{\R^n}^{}\Bigg[(G_{p}+k)^{p}-\frac{p-1}{2p}\;\mbox{\rm div}\;
   ((G_{p}+k)^{p-2}D(G_{p}+k))}\\
\\
\ds{-\frac{p-2}{p^{2}}\;\mbox{\rm div}\;((G_{p}+k)^{p-1}DU)
   \Bigg]\;|v|^{p}dx =\int_{\R^n}^{}g\;(G_{p}+k)^{p-1}|v|^{p-2}vdx.
   }
    \end{array}
   \end{equation}

   \end{Lemma}
   {\bf Proof}---We will assume for simplicity that $k=0.$
Multiplying both sides of (\ref{e1.4}) by
   $G_{p}^{p-1}|v|^{p-2}v,$ we obtain
   $$ \begin{array}{l}
   \ds{\lambda
   \int_{\R^n}^{}G_{p}^{p-1}|v|^{p}dx}-\ds{\frac{1}{2}\;\int_{\R^n}^{}\Delta v
   |v|^{p-2}vG_{p}^{p-1}dx
   }\\
   \\
   +\ds{ \frac{p-2}{p}\;\int_{\R^n} \langle DU,DG_{p} \rangle
G_{p}^{p-1}|v|^{p-2}v )dx+ \int_{\R^n}G_{p}^{p}|v|^{p}dx
   }\\
   \\
   =\ds{\int_{\R^n}^{}g\;G_{p}^{p-1}|v|^{p-2}vdx.
   }
    \end{array}
   $$
   An integration by parts in the second integral of the left hand side gives
   \begin{equation}
   \label{e2.2}
   \begin{array}{l}
   \ds{\lambda

\int_{\R^n}^{}G_{p}^{p-1}|v|^{p}dx}+\ds{\frac{p-1}{2}\;\int_{\R^n}^{}|Dv|^{2}|v|^{p-2}G_{p}^{p-1}dx
   }\\
   \\
+ \ds{\frac{p-2}{p}\;\int_{\R^n} \langle DU,DG_{p} \rangle
G_{p}^{p-1}|v|^{p-2}v )dx+ \int_{\R^n}G_{p}^{p}|v|^{p}dx
   }\\
   \\
  = \ds{\int_{\R^n}^{}gG_{p}^{p-1}|v|^{p-2}vdx.
   }
    \end{array}
   \end{equation}
  Now  two other  integrations by parts give
  %
  % equation
  %
  \begin{equation}
  \label{e2.3}
  \frac{p-2}{p}\; \int_{\R^n}\langle DU,DG_{p} \rangle G_{p}^{p-1}|v|^{p-2}v dx=
  -\frac{p-1}{2p}\;\int_{\R^n}^{}  \;\mbox{\rm div}\;
(G_{p}^{p-2}DG_{p}) \;|v|^{p}dx,
  \end{equation}
  %
  % equation
  %
  \begin{equation}
  \label{e2.4}
   \frac{p-2}{p}\; \int_{\R^n}\langle DU,DG_{p} \rangle
G_{p}^{p-1}|v|^{p-2}v dx=
   -\frac{p-2}{p^{2}}\; \int_{\R^n}^{} \;\mbox{\rm div}\;  ( G_{p}^{p-1}DU)
   \;|v|^{p}dx.
  \end{equation}
  Substituting (\ref{e2.3}) and (\ref{e2.4}) in (\ref{e2.2})  gives the
  conclusion.\qed

   From Lemma \ref{l2.4} it follows that we can estimate
  the $L^{p}(\R^n,dx)$ norm of $G_p v$ provided the following assumption holds
  \begin{Hypothesis}
  \label{h2.5}
  There exists $\alpha \in [0,1)$ such that
  %
  % equation
  %
  \begin{equation}
  \label{e2.5}
\begin{array}{l}
  \ds{(G_{p}+k)^{p}-\frac{p-1}{2p}\;\mbox{\rm div}\;
   ((G_{p}+k)^{p-2}DG_{p})} \\
\\
  \ds{-\frac{p-2}{p^{2}}\;\mbox{\rm div}\;((G_{p}+k)^{p-1}DU)\le
\alpha (G_{p}+k)^{p}.}
\end{array}
  \end{equation}

  \end{Hypothesis}
  Notice that this assumption is   fulfilled in the   Example
  \ref{Ex2.7}.

  We can prove finally the result
  \begin{Theorem}
  \label{t2.6}
  Assume that Hypotheses \ref{h1.1}, \ref{h1.3}--(i)--(iii), \ref{h2.1}
  and \ref{h2.5} holds and that $p\ge 2$. 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}---Let $v$ be a variational solution of equation
\ref{e1.4}.  By Lemma
  \ref{l2.4} we get that
  %
  % displaymath
  %
  \begin{displaymath}
   (1-\alpha )\int_{\R^n}^{} (G_{p}+k)^{p}|v|^{p}dx \le
   \int_{\R^n}^{}|g|\;(G_{p}+k)^{p-1}|v|^{p-1}dx.
  \end{displaymath}
  Consequently, by the H\"older inequality,
  %
  % displaymath
  %
  \begin{displaymath}
  \int_{\R^n}^{} (G_{p}+k)^{p}|v|^{p}dx\le \left( \int_{\R^n}^{}|g|^{p}dx
  \right)^{ \frac{1}{p}}\left(\int_{\R^n}^{} (G_{p}+k)^{p}|v|^{p}dx
  \right)^{\frac{p-1}{p}},
  \end{displaymath}
  and so
  %
  % displaymath
  %
  \begin{displaymath}
  \int_{\R^n}^{}|g|^{p}dx\le (1-\alpha )^{-p}\int_{\R^n}^{}|g|^{p}dx.
  \end{displaymath}
  Therefore
  there exists     $c>0$ 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

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