On functions of bounded variation on convex domains in Hilbert spaces

Abstract

We study functions of bounded variation (and sets of finite perimeter) on a convex open set ${\varOmega }\subseteq X$, X being an infinite-dimensional separable real Hilbert space. We relate the total variation of such functions, defined through an integration by parts formula, to the short-time behaviour of the semigroup associated with a perturbation of the Ornstein–Uhlenbeck operator.

1 Introduction

In this paper we study some properties of functions of bounded variation (BV functions, for short) defined on an open convex subset of a real separable Hilbert space, endowed with a weighted Gaussian measure.

In finite dimension the theory of BV functions is widely developed (see e.g. [3] and the references therein), whereas in the infinite-dimensional setting the analysis is still at the initial stage and many basic properties are unexplored. Besides the interest on its own, the study of BV functions in infinite-dimensional spaces is motivated by problems arising in calculus of variations, stochastic analysis and it is connected with the applications in information technology (see, for example, [19, 22,23,24, 26]).

BV functions for Gaussian measures in separable Banach spaces were introduced in [17] using Dirichlet forms. Inspired by the results in finite dimension, which connect the theory of functions of bounded variation to that of semigroups of bounded operators, the authors of [18] have proved an elegant characterisation of BV functions in terms of the short-time behaviour of the Ornstein–Uhlenbeck semigroup. More precisely, in a separable Banach space X, if $\gamma$ is a centred and nondegenerate Gaussian measure on X and u belongs to the Orlicz space $L(\log L)^{1/2}(X,\gamma )$, then $u\in BV(X, \gamma )$ if, and only if,

$$\begin{aligned} \liminf _{t \rightarrow 0^+}\int _X |D_HS(t)u|_H\mathrm{d}\gamma <+\infty , \end{aligned}$$

where $D_H$ is the gradient operator along the Cameron–Martin space H (see Sect. 2) and S(t) is the classical Ornstein–Uhlenbeck semigroup defined via the Mehler formula (see (1)). The latter is the analogous, in the Gaussian setting, of the heat semigroup used by De Giorgi in [11] to provide the original definition of BV functions in the Euclidean case. An analytic approach based on geometric measure theory is proposed in [4] to prove, as in the finite-dimensional case, the equivalence of different definitions of $BV(X,\gamma )$ functions also, as in [18], in terms of the Ornstein–Uhlenbeck semigroup S(t) near $t=0$. Similar De Giorgi-type characterisations of BV functions have been obtained for weighted Gaussian measures and more recently for general Fomin differentiable measures in Hilbert spaces, see [12] and the reference therein.

Beside the difficulty of considering general measures, another difficulty of different nature comes from the consideration of functions defined in domains rather than in the whole space. These difficulties come from the lack of factorisation of the underlying measure (that is lost even for Gaussian measures in domains) and the unavailability of decomposition of the domain through the classical method of local charts. Therefore, the easiest interesting case seems to be that of convex domains that are possible to deal with through global penalisation techniques. This is the approach we followed in [6] (see also [21]), and in this paper we take advantage of the results proved there. We start from a weighted Gaussian measure $\nu :=e^{-U}\gamma$ in a Hilbert space X, where $U:X\rightarrow {{\mathbb {R}}}$ is convex and sufficiently regular, and consider an open convex domain $\varOmega \subseteq X$. After introducing the Cameron–Martin space H and the Malliavin gradient $D_H$ along it, we define the form $(u,v)\mapsto \int _\varOmega \langle D_Hu,D_Hv\rangle _H\mathrm{d}\nu$ on the appropriate Sobolev spaces. The perturbed Ornstein–Uhlenbeck operator $L_\varOmega$ is then defined in the usual variational way, and it is the generator of an analytic, strongly continuous and contraction semigroup $T_\varOmega (t)$ in $L^p(\varOmega ,\nu )$, for $1<p<\infty$.

For the latter, differently from the Ornstein–Uhlenbeck semigroup in the whole space, no explicit integral representation which allows for direct computations is known. In this direction, in [21] the authors consider the restrictions to an open convex set ${\varOmega }\subseteq X$ of $BV(X,\gamma )$ functions and they characterise the finiteness of their total variation in ${\varOmega }$ in terms of the Neumann Ornstein–Uhlenbeck semigroup defined in ${\varOmega }$.

Following the ideas in [2], we define the $BV(\varOmega ,\nu )$ space through an integration by parts formula against suitable Lipschitz functions. Then, we show that the functions u of bounded variation in ${\varOmega }$ with respect to $\nu$ can be characterised by the finiteness of the limit of $\Vert D_HT_\varOmega (t)u\Vert _{L^1(\varOmega ,\nu ;H)}$ as $t\rightarrow 0^+$. The proof of this result relies on a commutation formula between the semigroup $T_{\varOmega }(t)$ and the gradient operator along H (see Proposition 5). This result was already known in the case of the whole space (see [12]). Here, by means of the crucial pointwise gradient estimate (7) and suitable penalisations $\varPhi _\varepsilon$ of U outside $\varOmega$ based on the distance function from $\varOmega$ along H (here is a first point where the convexity of $\varOmega$ comes into the play) and the penalisation $\nu _\varepsilon :=e^{-\varPhi _\varepsilon }\gamma$ of the measure $\nu$, see Sect. 2.1, we are able to let $\varepsilon$ to $0^+$ and to come back to $\varOmega$.

Finally, we provide a necessary condition in order that a set E is of finite perimeter in ${\varOmega }$ with respect to $\nu$ (i.e. $\chi _E\in BV(\varOmega ,\nu )$). This condition is given in terms of the short-time behaviour of the Ornstein–Uhlenbeck content $\Vert T_\varOmega (t)\chi _E-\chi _E\Vert _{L^1(\varOmega ,\nu )}$ as $t\rightarrow 0^+$. Further, a sufficient condition in terms of a related quantity is also shown. This circle of ideas goes back to [20], which originated several researches. Among these, the only infinite-dimensional result, proved for BV functions in space endowed with a Gaussian measure, is in [5].

2 Hypotheses and preliminaries

Let $H_1$ and $H_2$ be two real Hilbert spaces with inner products ${\left\langle \cdot ,\cdot \right\rangle }_{H_1}$ and ${\left\langle \cdot ,\cdot \right\rangle }_{H_2},$ respectively. We denote by ${\mathcal {B}}(H_1)$ the $\sigma$-algebra of Borel subsets of $H_1$ and by $C^k_b(H_1;H_2)$, $k\in {{\mathbb {N}}}\cup \{\infty \}$ the set of k-times Fréchet differentiable functions from $H_1$ to $H_2$ with bounded derivatives up to order k ($C_b^k(H_1)$ if $H_2={{\mathbb {R}}}$). For $\varPhi \in C_b^1(H_1;H_2)$ we denote by ${{D}} \varPhi (x)$ the derivative of $\varPhi$ at $x\in H_1$: if $f\in C_b^1(H_1)$, for every $x\in H_1$ there exists a unique $k\in H_1$ such that $D f(x)(h)={\left\langle h,k\right\rangle }_{H_1}$, $h\in H_1$ and we set $D f(x):=k$. Let X be a separable Hilbert space, with inner product $\langle \cdot ,\cdot \rangle$ and norm $|\cdot |$. Let $B\in {\mathcal {L}}(X)$ (the set of bounded linear operators from X to itself). We say that B is non-negative if ${\left\langle Bx,x\right\rangle }\ge 0$ for every $x\in X$ and positive if ${\left\langle Bx,x\right\rangle }>0$ for every $x\in X\setminus {\left\{ 0\right\} }$. We recall that a non-negative and self-adjoint operator $B\in {\mathcal {L}}(X)$ is a trace class operator whenever $\mathrm{Tr}(B):=\sum _{n=1}^{\infty }\langle Be_n,e_n\rangle <\infty$ for some (and hence, every) orthonormal basis $(e_n)_{n\in {{\mathbb {N}}}}$ of X.

Let $\gamma$ be a nondegenerate Gaussian measure on X with mean zero and covariance operator $Q_\infty :=-QA^{-1}$, where the operators Q and A satisfy the following assumptions.

Hypotheses 1

• (i) $Q\in {\mathcal {L}}(X)$ is a self-adjoint and non-negative operator with $\mathrm{Ker}\, Q=\{0\}$;

• (ii) $A:D(A)\subseteq X\rightarrow X$ is a self-adjoint operator satisfying ${\left\langle Ax,x\right\rangle }\le -\omega {\left| x\right| }^2$ for every $x\in D(A)$ and some positive $\omega$;

• (iii) $Qe^{tA}=e^{tA}Q$ for any $t\ge 0$;

• (iv) $\mathrm{Tr}(-QA^{-1})<\infty$.

Under Hypotheses 1(i)–(iii), the measure $\gamma$ is well defined and the Ornstein–Uhlenbeck semigroup defined via the Mehler formula

$$\begin{aligned} (S(t)f)(x):=\int _Xf(e^{-t}x+\sqrt{1-e^{-2t}}y)\mathrm{d}\gamma (y),\qquad x\in X,\,f\in L^1(X,\gamma ) \end{aligned}$$

(1)

is symmetric in $L^2(X,\gamma )$. We fix an orthonormal basis $(v_k)_{k\in {{\mathbb {N}}}}$ of X such that

$$\begin{aligned} Q_\infty v_k=\lambda _k v_k,\qquad k\in {{\mathbb {N}}}, \end{aligned}$$

(2)

where $(\lambda _k)_{k\in {{\mathbb {N}}}}$ is the decreasing sequence of eigenvalues of $Q_\infty$. Under Hypothesis 1(iv), the Cameron–Martin space $(H, {\left| \cdot \right| }_H)$

$$\begin{aligned} H:=Q_\infty ^{1/2}(X)=\Bigl \{x\in X\Big | \sum _{k=1}^{\infty }\lambda _k^{-1}{\left\langle x,v_k\right\rangle }^2<\infty \Bigr \}, \end{aligned}$$

where ${\left| \cdot \right| }_H$ is induced by the inner product ${\left\langle h,k\right\rangle }_H:=\langle Q_\infty ^{-1/2}h,Q_\infty ^{-1/2}k\rangle$, is a Hilbert space compactly and densely embedded in X (see [8] and [13] for further details). The sequence $(e_k)_{k\in {{\mathbb {N}}}}$, where $e_k=\sqrt{\lambda _k}v_k$ for any $k\in {{\mathbb {N}}}$, is an orthonormal basis of H. By Hypotheses 1, the operator $-Q^{-1}_\infty :D(Q^{-1}_\infty )\subseteq X\rightarrow X$ ($-Q^{-1}_\infty :D(Q^{-1}_\infty )\subseteq H\rightarrow H$, respectively) is the generator of a contractive and strongly continuous semigroup $e^{-tQ_\infty ^{-1}}$ on X (on H, respectively), see [14, Proposition p. 84]). If Y is a Banach space with norm ${\left\| \cdot \right\| }_Y$, a function $F:X\rightarrow Y$ is said to be H-Lipschitz continuous if there exists a positive constant C such that

$$\begin{aligned} {\left\| F(x+h)-F(x)\right\| }_Y\le C{\left| h\right| }_H, \end{aligned}$$

(3)

for every $h\in H$ and $\gamma$-a.e. $x\in X$. We denote by $[F]_{H\text {-Lip}}$ the best constant C in (3). For more information see [8, Sections 4.5 and 5.11]. We denote by ${\mathcal {H}}_2$ the space of the Hilbert–Schmidt operators in H that is the space of the bounded linear operators $B:H\rightarrow H$ such that ${\left\| B\right\| }_{{\mathcal {H}}_2}^2:=\sum _{i=1}^{\infty }{\left| Bg_i\right| }^2_H$ is finite, where $\{g_n\,|\,n\in {{\mathbb {N}}}\}$ is any orthonormal basis of H. We say that $f:X\rightarrow {{\mathbb {R}}}$ is H-differentiable at $x_0\in X$ if there exists $\ell \in H$ such that

$$\begin{aligned} f(x_0+h)=f(x_0)+{\left\langle \ell ,h\right\rangle }_H+o(|h|_H),\qquad \text {as} \ \ |h|_H\rightarrow 0. \end{aligned}$$

In such a case we set $D_H f(x_0):=\ell$ and $D_i f(x_0):= \langle D_H f(x_0), e_i\rangle _H$ for any $i\in {{\mathbb {N}}}$. The derivative $D_H f(x_0)$ is called the Malliavin derivative of f at $x_0$. In a similar way we say that f is twice H-differentiable at $x_0$ if f is H-differentiable near $x_0$ and there exists ${\mathcal {B}}\in {\mathcal {H}}_2$ such that

$$\begin{aligned} f(x_0+h)=f(x_0)+{\left\langle D_H f(x_0),h\right\rangle }_H+\frac{1}{2}\langle {\mathcal {B}} h,h\rangle _H+o(|h|^2_H),\qquad \text {as} \ \ |h|_H\rightarrow 0. \end{aligned}$$

In such a case we set $D^2_H f(x_0):={\mathcal {B}}$ and $D_{ij} f(x_0):= \langle D^2_H f(x_0)e_j, e_i\rangle _H$ for any $i,j\in {{\mathbb {N}}}$. If f is twice H-differentiable at $x_0$, then $D_{ij}f(x_0)=D_{ji}f(x_0)$ for every $i,j\in {{\mathbb {N}}}$. Notice that if $f:X\rightarrow {{\mathbb {R}}}$ is once or twice Fréchet differentiable at $x_0$, then it is once or twice H-differentiable at $x_0$ and it holds $D_Hf(x_0)=Q_{\infty }Df(x_0)$, and $D^2_Hf(x_0)=Q_{\infty }D^2 f(x_0)Q_{\infty }$, where the equality must be understood as holding in H. For any $k\in {{\mathbb {N}}}\cup {\left\{ \infty \right\} }$, we denote by ${\mathcal {F}}C_b^k(X)$, the space of cylindrical $C^k_b$ functions, i.e. the set of functions $f:X\rightarrow {{\mathbb {R}}}$ such that $f(x)=\varphi (\langle x, h_1\rangle ,\ldots , \langle x, h_N\rangle )$ for some $\varphi \in C_b^k({{\mathbb {R}}}^N)$, $h_1,\ldots , h_N\in H$ and $N\in {{\mathbb {N}}}$. By ${\mathcal {F}}C_b^k(X,H)$ we denote H-valued cylindrical $C_b^k$ functions with finite rank. The Sobolev spaces in the sense of Malliavin $D^{1,p}(X,\gamma )$ and $D^{2,p}(X,\gamma )$ with $p\in [1,\infty )$ are defined as the completions of the smooth cylindrical functions ${\mathcal {F}}C_b^\infty (X)$ in the norms

$$\begin{aligned} {\left\| f\right\| }_{D^{1,p}(X,\gamma )}:= & {} \Bigl ({\left\| f\right\| }^p_{L^p(X,\gamma )}+\int _X{\left| D_H f\right| }_H^p\mathrm{d}\gamma \Bigr )^{\frac{1}{p}}; \\ {\left\| f\right\| }_{D^{2,p}(X,\gamma )}:= & {} \Bigl ({\left\| f\right\| }^p_{D^{1,p}(X,\gamma )}+\int _X\Vert D_H^2 f\Vert ^p_{{\mathcal {H}}_2}\mathrm{d}\gamma \Bigr )^{\frac{1}{p}}. \end{aligned}$$

This is equivalent to considering the domain of the closure of the gradient operator, defined on smooth cylindrical functions, in $L^p(X,\gamma )$ (see [8, Section 5.2]). Let $U:X\rightarrow {{\mathbb {R}}}$ satisfy the following assumptions.

Hypotheses 2

U is a convex function which belongs to $C^2(X)\cap D^{1,q}(X,\gamma )$ for all $q\in [1,\infty )$ with H-Lipschitz gradient.

The convexity of the function U guarantees that U is bounded from below by a linear function, and therefore, it decreases at most linearly and by Fernique theorem (see [8, Theorem 2.8.5]) $e^{-U}$ belongs to $L^1(X,\gamma )$. Then, we can consider the finite log-concave measure

$$\begin{aligned} \nu := e^{-U}\gamma . \end{aligned}$$

It is obvious that $\gamma$ and $\nu$ are equivalent measures, hence saying that a statement holds $\gamma$-a.e. is the same as saying that it holds $\nu$-a.e. Moreover as $U\in \cap _{q\ge 1}D^{1,q}(X,\gamma )$, the operator $D_H:{\mathcal {F}}C^1_b(X)\rightarrow L^p(X,\nu ;H)$ is closable in $L^p(X,\nu )$, $p\in (1,\infty )$ and the space $D^{1,p}(X,\nu )$, $p>1$ can be defined as the domain of its closure (still denoted by $D_H$). In a similar way we may define $D^{2,p}(X,\nu )$, $p\in (1,\infty )$ (for more details see [1, 9, 16]). The Gaussian integration by parts formula $\int _X D_i f \mathrm{d}\gamma =\frac{1}{\sqrt{\lambda _i}}\int _X \langle x,v_i\rangle f \mathrm{d}\gamma$, which holds true for any $f\in {\mathcal {F}}C^1_b(X)$ and $i\in {{\mathbb {N}}}$, yields

$$\begin{aligned} \int _X \psi D_i \varphi \mathrm{d}\nu +\int _X\varphi D_i\psi \mathrm{d}\nu =\int _X \varphi \psi D_i U \mathrm{d}\nu +\frac{1}{\sqrt{\lambda _i}} \int _X \langle x, v_i\rangle \varphi \psi \mathrm{d}\nu ,\qquad \;\,i \in {{\mathbb {N}}}, \end{aligned}$$

for any $\varphi \in D^{1,p}(X,\nu )$ ($p>1$) and $\psi \in {\mathcal {F}}C^1_b(X)$.

In what follows $\varOmega$ denotes an open subset of X. In this case, the spaces $D^{1,p}(\varOmega ,\nu )$ and $D^{2,p}(\varOmega ,\nu )$, $p\in (1,\infty )$, can be defined in a similar way as in the whole space, thanks to the following result (see [6, Proposition 1.4]).

Proposition 1

Assume that Hypotheses 1 and 2 are satisfied. Let $p\in (1,\infty )$ and let ${\varOmega }$ be an open subset of X. The operators $D_H:{\mathcal {F}}C_b^\infty (\varOmega )\rightarrow L^p(\varOmega ,\nu ; H)$ and

$$\begin{aligned} (D_H,D_H^2):{\mathcal {F}}C_b^\infty (\varOmega )\times {\mathcal {F}}C_b^\infty (\varOmega )\rightarrow L^p(\varOmega ,\nu ; H)\times L^p(\varOmega ,\nu ; {\mathcal {H}}_2) \end{aligned}$$

are closable in $L^p(\varOmega ,\nu )$ and $L^p(\varOmega ,\nu )\times L^p(\varOmega ,\nu )$, respectively. Here ${\mathcal {F}}C_b^\infty (\varOmega )$ is the space of the restrictions to $\varOmega$ of functions in ${\mathcal {F}}C_b^\infty (X)$.

The spaces $D^{1,p}(\varOmega ,\nu ;H)$, $p\in (1,\infty )$, are defined in a similar way, replacing smooth cylindrical functions with H-valued smooth cylindrical functions with finite rank. We recall that if $F\in D^{1,p}(\varOmega ,\nu ;H)$, then $D_H F(x)$ belongs to ${\mathcal {H}}_2$ for a.e. $x\in {\varOmega }$. We denote by $p'$ the conjugate exponent to $p\in (1,\infty )$.

2.1 Perturbed Ornstein–Uhlenbeck semigroup on convex domains

In order to consider the initial boundary value problems defined in ${\varOmega }$ we define the distance function along H

$$\begin{aligned} d_{\varOmega }(x):=\left\{ \begin{array}{lc} \inf \{|h|_H\,|\, h\in H\cap (\varOmega -x)\}, \quad \quad &{} H\cap (\varOmega -x)\ne \emptyset ;\\ \infty , &{} H\cap (\varOmega -x)=\emptyset , \end{array}\right. \end{aligned}$$

for $\ x\in X$, and we recall some useful regularity results, (see, for instance, [8, Theorems 2.8.5 and 5.11.2] and [10, Section 3]).

Proposition 2

If $\varOmega \subseteq X$ is an open convex set, then $d_\varOmega ^2$ is H-differentiable and its Malliavin derivative is H-Lipschitz with H-Lipschitz constant less than or equal to 2, i.e.

$$\begin{aligned}\ |D_Hd_{\varOmega }^2(x+h)-D_Hd_{{\varOmega }}^2(x)|_H\le 2 |h|_H, \end{aligned}$$

for any $h\in H$ and for $\nu$-a.e $x\in X$. Moreover, $D_H^2d_\varOmega ^2$ exists $\nu$-a.e. in X and $d_\varOmega ^2$ belongs to $D^{2,p}(X,\nu )$ for every $p\in [1,\infty )$.

We require some further regularity on $d_\varOmega ^2$.

Hypotheses 3

Let ${\varOmega }$ be an open convex subset of X such that $\nu (\partial \varOmega )=0$ and $D_H^2d_\varOmega ^2$ is H-continuous $\gamma$-a.e. in X, i.e. for $\gamma$-a.e. $x\in X$ we have

$$\begin{aligned} \lim _{{\left| h\right| }_H\rightarrow 0}D^2_H d^2_\varOmega (x+h)=D^2_H d^2_\varOmega (x). \end{aligned}$$

Remark 1

As stated in [6, Remark 1.7] there is a rather large class of subsets of X satisfying Hypothesis 3. For instance, if $\partial \varOmega$ is (locally) a $C^2$-embedding in X of an open subset of a hyperplane in X and $\nu (\partial \varOmega )=0$, then Hypothesis 3 is satisfied. Easy examples are open balls and open ellipsoids of X, open hyperplanes of X and every set of the form $\varOmega ={\left\{ x\in X\, |\,G(x)< 0\right\} }$, where $G:X\rightarrow {{\mathbb {R}}}$ is a $C^2$-convex function such that $D_H G$ is nonzero at every point of $\partial \varOmega$.

We consider the semigroup $T_{\varOmega }(t)$ on $L^2({\varOmega },\nu )$ and its generator $L_\varOmega$:

$$\begin{aligned}&D(L_{\varOmega })=\Bigl \{u\in D^{1,2}(\varOmega ,\nu )\,\Big |\, \exists v\in L^2(\varOmega ,\nu )\text { such that } \nonumber \\&\quad \int _\varOmega \langle D_Hu,D_H\varphi \rangle _H\mathrm{d}\nu =-\int _\varOmega v\varphi \, \mathrm{d}\nu \ \forall \varphi \in {\mathcal {F}}C^\infty _b({\varOmega })\Bigr \} \end{aligned}$$

(4)

with $L_{\varOmega }u:=v$ if $u\in D(L_{\varOmega })$. We recall (see [6, Section 2]) an approximation procedure of $T_{\varOmega }(t)f$, when $f\in L^2({\varOmega }, \nu )$, through ${\mathcal {F}}C^3_b(X)$ functions that relies on a reduction to a finite (say n-) dimensional space and on a $\varepsilon$-penalisation argument. Accordingly, the approximation depends on two parameters n and $\varepsilon$. More precisely, we consider the function $\varPhi _\varepsilon :X\rightarrow {{\mathbb {R}}}$ defined by

$$\begin{aligned} \varPhi _\varepsilon (x):=U(x)+\frac{1}{2\varepsilon }d^2_\varOmega (x),\qquad \;\, x\in X,\, \varepsilon >0, \end{aligned}$$

and the measure $\nu _\varepsilon$ given by $e^{-\varPhi _\varepsilon }\gamma$. Next, we consider the operator $L_\varepsilon$ on the whole X defined as

$$\begin{aligned}&D(L_{\varepsilon })=\Bigl \{u\in D^{1,2}(X,\nu _\varepsilon )\,\Big |\, \exists \ v\in L^2(X,\nu _\varepsilon )\text { such that } \nonumber \\&\quad \int _X{\left\langle D_H u,D_H \varphi \right\rangle }_H\mathrm{d}\nu _\varepsilon =-\int _X v\varphi \, \mathrm{d}\nu _\varepsilon \text { for every }\varphi \in {\mathcal {F}}C^\infty _b(X)\Bigr \}, \end{aligned}$$

(5)

with $L_{\varepsilon }u:=v$ if $u\in D(L_{\varepsilon })$, and the semigroup $T_\varepsilon (t)$ generated by $L_\varepsilon$ in $L^2(X, \nu _\varepsilon )$. We point out that $L_\varepsilon$ acts on smooth cylindrical functions $\varphi$ as follows

$$\begin{aligned} L_\varepsilon \varphi&=\mathrm{Tr}(D^2_H \varphi )-\sum _{i=1}^{\infty }\lambda _i^{-1}\langle x,e_i\rangle D_i\varphi -\langle D_H\varPhi _\varepsilon ,D_H\varphi \rangle _H\\&=\mathrm{Tr}(D^2_H \varphi )-\sum _{i=1}^{\infty }\lambda _i^{-1}\langle x,e_i\rangle D_i\varphi -\Big \langle D_HU+\frac{1}{2\varepsilon }D_H d^2_{\varOmega },D_H\varphi \Big \rangle _H. \end{aligned}$$

Now we recall a useful approximation result whose proof can be found in [6, Theorem 2.8].

Theorem 1

Under Hypotheses 1, 2 and 3 the following statements hold true.

• (i) For any $\varepsilon >0$ and $f\in L^2(X,\nu _\varepsilon )$, there exists a sequence $(f_n)_{n\in {{\mathbb {N}}}}\subseteq L^2(X,\nu _\varepsilon )$ converging to f in $L^2(X, \nu _\varepsilon )$ such that $T_\varepsilon (t)f_n$ is in ${\mathcal {F}}C^3_b(X)$ and

  $$\begin{aligned} \lim _{n\rightarrow \infty }{\left\| T_{\varepsilon }(t)f_n-T_\varepsilon (t) f\right\| }_{D^{1,2}(X,\nu _\varepsilon )}=0,\qquad \;\, t>0. \end{aligned}$$

  In addition, if $f\in D^{1,2}(X,\nu _\varepsilon )$ then the sequence $(f_n)$ can be chosen in a way that $D_Hf_n$ converges to $D_Hf$ in $L^1(X, \nu _\varepsilon ; H)$, as $n\rightarrow \infty$.

• (i) For any $f\in L^2(\varOmega ,\nu )$ there exists an infinitesimal sequence $(\varepsilon _n)_{n\in {{\mathbb {N}}}}$ such that $T_{\varepsilon _n}(t) {\widetilde{f}}$ weakly converges to $T_\varOmega (t) f$ in $D^{1,2}(\varOmega ,\nu )$, where ${\widetilde{f}}$ is any $L^2$-extension of f to X.

We collect some properties of $T_{\varOmega }(t)$, see [6, Proposition 1.10, Theorems 3.1 & 3.3].

Proposition 3

If Hypotheses 1, 2 and 3 hold true, then

• (i) the semigroup $T_\varOmega (t)$ generated in $L^2({\varOmega },\nu )$ can be extended to a positivity preserving contraction semigroup in $L^p(\varOmega ,\nu )$ for every $1\le p\le \infty$ and $t\ge 0$, still denoted by $T_\varOmega (t)$. It is strongly continuous in $L^p({\varOmega }, \nu )$ for any $p \in [1,\infty )$ and consistent;

• (ii) for any $p\in [1,\infty )$, $f\in L^p(\varOmega ,\nu )$ and $g\in L^\infty (\varOmega ,\nu )$ it holds

  $$\begin{aligned} \int _\varOmega f T_\varOmega (t)g\mathrm{d}\nu =\int _\varOmega gT_\varOmega (t)f\mathrm{d}\nu ,\qquad t>0; \end{aligned}$$

  (6)

• (iii) for any $p\in (1,\infty )$, $f\in L^p({\varOmega },\nu )$ and $t>0$ there is $K_p>0$ such that

  $$\begin{aligned} |D_HT_{\varOmega }(t)f|_H^p\le K_pt^{-p/2}T_{\varOmega }(t)|f|^p\qquad \nu \text {-a.e. in }{\varOmega }; \end{aligned}$$

  (7)

• (iv) if $f\in D^{1,p}({\varOmega },\nu )$, $t>0$ and $p\in [1,\infty )$ it holds

  $$\begin{aligned} |D_H T_{\varOmega }(t) f|^p\le e^{-p\lambda _1^{-1}t}T_{\varOmega }(t)|D_H f|^p_H\qquad \nu \text {-a.e. in }{\varOmega }. \end{aligned}$$

  (8)

We point out that the results in Proposition 3 continue to hold if we replace ${\varOmega }$, $\nu$ and $T_{\varOmega }(t)$ by X, $\nu _\varepsilon$ and $T_\varepsilon (t)$, respectively.

2.2 BV functions in Hilbert spaces: definitions and some known facts

We introduce BV functions in the Wiener space setting. Let Y be a separable Hilbert space with norm ${\left| \cdot \right| }_Y$. We recall that in a separable space X the $\sigma$-algebra ${\mathcal {B}}(X)$ is generated by the family of the cylindrical sets (see e.g. [25]). Denote by ${\mathcal {M}}({\varOmega };Y)$ the set of Borel Y-valued measures on ${\varOmega }$. If $Y={{\mathbb {R}}}$ then we write ${\mathcal {M}}({\varOmega })$. The total variation of $\mu \in {\mathcal {M}}({\varOmega };Y)$ is the positive Borel measure

$$\begin{aligned} |\mu |(B):=\sup {\left\{ \sum _{n=1}^{\infty }|\mu (B_n)|_Y\, |\,\begin{array}{c} B=\bigcup _{n=1}^{\infty }B_n,\ B_n \in {\mathcal {B}}({\varOmega })\\ B_n\cap B_m=\emptyset , \text { if } n\ne m, \end{array}\right\} },\ \ B\in {\mathcal {B}}(\varOmega ). \end{aligned}$$

Let $\mathrm{Lip}_c(\varOmega ;Y)$ be the set of bounded Lipschitz continuous Y-valued functions $g:\varOmega \rightarrow Y$ such that $\mathrm{dist}(\mathrm{supp}\, g,X\smallsetminus \varOmega )>0$ and define the space $BV({\varOmega },\nu )$ as follows.

Definition 1

Let $\varOmega$ be an open subset of X. We say that a function $f\in L^2(\varOmega ,\nu )$ is of bounded variation in $\varOmega$, and we write $f\in BV(\varOmega ,\nu )$, if there exists a measure $\mu \in {\mathcal {M}}(\varOmega ;H)$ such that

$$\begin{aligned} \int _\varOmega f\partial _h^* g\mathrm{d}\nu =-\int _\varOmega gd{\left\langle \mu ,h\right\rangle }_H, \end{aligned}$$

for every $g\in \mathrm{Lip}_c(\varOmega )$ and $h\in H$, where $\partial _h^*$ denotes, up to the sign, the adjoint in $L^2({\varOmega },\nu )$ of the partial derivative along $h\in H$. In this case we set $D_\nu f:=\mu$.

As in the finite-dimensional case, one can characterise functions of bounded variation by their total variation.

Definition 2

Let $\varOmega$ be an open subset of X and $u\in L^2(\varOmega ,\nu )$. We define the variation of u in $\varOmega$ by

Here $\mathrm{div}_\nu \, g= \sum _{i=1}^N \partial _{k_i}^*g_i(x)$ if $g(x)=\sum _{i=1}^N g_i(x)k_i$ and $F=\mathrm{span}\{k_1,\ldots , k_N\}$ for some $N\in {{\mathbb {N}}}$.

When ${\varOmega }=X$, in the two definitions above we can consider $\mathrm{Lip}_b(X)$ and $\mathrm{Lip}_b(X;F)$, respectively, as test functions spaces.

As announced, in [2, Theorem 5.7] it has been proved that $u\in BV(\varOmega ,\nu )$ if and only if $V_\nu (u,\varOmega )$ is finite. Moreover, in this case

$$\begin{aligned} |D_\nu u|(\varOmega )=V_\nu (u,\varOmega ). \end{aligned}$$

(9)

Finally we say that a Borel subset E of X is of finite perimeter in $\varOmega$ with respect to $\nu$, whenever the function $\chi _E$ belongs to $BV({\varOmega },\nu )$. In this case we denote by $P_\nu (E,\varOmega )$ the total variation of $\chi _E$ in ${\varOmega }$.

3 A De Giorgi type characterisation

The main result of this section is the De Giorgi type characterisation of $BV({\varOmega }, \nu )$ functions in Theorem 3, which relies on a “quasi-commutative” formula between the semigroup $T_\varOmega (t)$ and the H-gradient operator $D_H$; here estimate (8) plays a crucial role. This formula is inspired by an analogous formula proved in [12]. We first define the Sobolev spaces $D^{1,2}(X,\nu _\varepsilon ;H)$.

Definition 3

We denote by $D^{1,2}(X,\nu _\varepsilon ;H)$ the domain of the closure of the operator $D_H:{\mathcal {F}}C_b^1(X,H)\rightarrow L^2(X,\nu _\varepsilon ;{\mathcal {H}}_2)$ in the $L^2(X,\nu _\varepsilon ;H)$ norm (see [7, Section 8.1]). $D_H$ is defined as

$$\begin{aligned} D_H\varPhi (x)=\sum _{i=1}^n\sum _{j=1}^{k(i)}\frac{\partial \varphi _i}{\partial \xi _j}(\langle x,x_1\rangle ,\ldots ,\langle x,x_{k(i)}\rangle )((Q^{1/2}_\infty x_j)\otimes e_i), \end{aligned}$$

where $\{e_i\,|\,i\in {{\mathbb {N}}}\}$ is an orthonormal basis of H and

$$\begin{aligned} \varPhi (x)=\sum _{i=1}^n\varphi _i(\langle x,x_1\rangle ,\ldots ,\langle x,x_{k(i)}\rangle ) e_i \end{aligned}$$

for some $n\in {{\mathbb {N}}}$, $k(i)\in {{\mathbb {N}}}$, $x_1,\ldots , x_{k(i)}\in X$ and $\varphi _i\in C_b^1({{\mathbb {R}}}^{k(i)})$ for every $i=1,\ldots ,n$. In an analogous way we define the space $D^{1,2}(\varOmega ,\nu ;H)$.

We first show a vector-valued version of Theorem 1. Let $\mathbf{L}_{\varepsilon }$ in $L^2(X,\nu _\varepsilon ;H)$ be the operator defined via the quadratic form by

$$\begin{aligned} (F,G)\mapsto \int _X{\left\langle D_H F,D_H G\right\rangle }_{{\mathcal {H}}_{\,2}}\mathrm{d}\nu _\varepsilon \qquad F,G\in D^{1,2}(X,\nu _\varepsilon ;H). \end{aligned}$$

In the same way we define the operator $\mathbf{L}_\varOmega$ in in $L^2({\varOmega },\nu ;H)$. We recall that by [14, p. 84] ( [14, Corollary 3.17 and Proposition 3.23] and [14, Corollary 4.8], respectively), the operators $\mathbf{L}_{\varepsilon }$ and $\mathbf{L}_{\varOmega }$ generate strongly continuous semigroups $\mathbf{T}_\varepsilon (t)$ and $\mathbf{T}_\varOmega (t)$ (contractive and analytic, respectively).

Proposition 4

The operators $\mathbf{L}_\varepsilon$, $\mathbf{L}_\varOmega$ and the semigroups $\mathbf{T}_\varepsilon (t)$ and $\mathbf{T}_\varOmega (t)$ act component by component, i.e. if $F\in D(\mathbf{L}_\varepsilon )$ $(D(\mathbf{L}_\varOmega )$, respectively), and it is such that $F=\sum _{i=1}^{\infty }f_ie_i$ for some basis $\{e_n\,|\,n\in {{\mathbb {N}}}\}$ of H, then $f_i\in D(L_\varepsilon )$ $(D(L_\varOmega )$, respectively) and

$$\begin{aligned} \mathbf{L}_\varepsilon F=\sum _{i=1}^{\infty }(L_\varepsilon f_i)e_i,\qquad \mathbf{L}_\varOmega F= \sum _{i=1}^{\infty }(L_\varOmega f_i)e_i . \end{aligned}$$

Moreover, for every $t>0$, if $F\in L^2(X,\nu _\varepsilon ;H)$ $(L^2(\varOmega ,\nu ;H)$, respectively), and it is such that $F=\sum _{i=1}^{\infty }f_ie_i$ for some basis $\{e_n\,|\,n\in {{\mathbb {N}}}\}$ of H and $f_i\in L^2(X,\nu _\varepsilon )$ ($L^2(\varOmega ,\nu )$, respectively) then

$$\begin{aligned} \mathbf{T}_\varepsilon (t)F=\sum _{i=1}^{\infty }(T_\varepsilon (t)f_i)e_i,\qquad {\left( \mathbf{T}_\varOmega (t)F= \sum _{i=1}^{\infty }(T_\varOmega (t)f_i)e_i\text {, respectively}\right) }. \end{aligned}$$

The above identities hold $\nu _\varepsilon$-a.e. in X ($\nu _{\varOmega }$-a.e. in ${\varOmega }$, respectively).

Proof

We only show the results for $\mathbf{L}_\varepsilon$ and $\mathbf{T}_\varepsilon (t)$. Let $F=\sum _{i=1}^{\infty }f_ie_i \in D(\mathbf{L}_\varepsilon )$ and let $G=ge_j$ for some $j\in {{\mathbb {N}}}$ and $g\in D^{1,2}(X,\nu _\varepsilon )$; then

$$\begin{aligned} \int _X{\left\langle D_H g,D_H f_j\right\rangle }_H\mathrm{d}\nu _\varepsilon&=\int _X{\left\langle D_H G,D_H F\right\rangle }_{{\mathcal {H}}_2}\mathrm{d}\nu _\varepsilon \\&=-\int _X{\left\langle G,\mathbf{L}_\varepsilon F\right\rangle }_H\mathrm{d}\nu _\varepsilon =-\int _X g(\mathbf{L}_\varepsilon F)_j\mathrm{d}\nu _\varepsilon . \end{aligned}$$

This shows that $f_j\in D(L_\varepsilon )$ (see (5)) and $L_\varepsilon f_j=(\mathbf{L}_\varepsilon F)_j$. Now observe that

$$\begin{aligned} D_t(\mathbf{T}_\varepsilon (t)F)_j=(\mathbf{L}_\varepsilon \mathbf{T}_\varepsilon (t)F)_j=L_\varepsilon (\mathbf{T}_\varepsilon (t)F)_j,\qquad (\mathbf{T}_\varepsilon (0)F)_j=f_j. \end{aligned}$$

Thus, by the uniqueness of the solution of the Cauchy problem associated with $D_t-L_\varepsilon$ in $L^2(X, \nu _\varepsilon )$, it follows that $(\mathbf{T}_\varepsilon (t)F)_j=T_\varepsilon (t)f_j$ for any $t>0$. The arbitrariness of $j\in {{\mathbb {N}}}$ concludes the proof. $\square$

Remark 2

According to the definition of $\mathbf{T}_\varepsilon (t)$ and $\mathbf{T}_\varOmega (t)$ it is immediately seen that for every $F\in L^2(X, \nu _\varepsilon ;H)$ and $G\in L^2({\varOmega },\nu ;H)$

$$\begin{aligned} |\mathbf{T}_\varepsilon (t)F|^2\le T_\varepsilon (t)|F|^2, \qquad t\ge 0,\ \nu _\varepsilon \text {-a.e. in } X \end{aligned}$$

(10)

and

$$\begin{aligned} |\mathbf{T}_{\varOmega }(t)G|^2\le T_{\varOmega }(t)|G|^2, \qquad t\ge 0,\ \nu \text {-a.e. in } {\varOmega }. \end{aligned}$$

(11)

Moreover, taking into account that the semigroups $\mathbf{T}_\varOmega (t)$ and $\mathbf{T}_\varepsilon (t)$ act component by component, we can obtain a vector-valued version of Theorem 1.

Theorem 2

Under Hypotheses 1, 2 and 3, the following statements hold true.

• (i) For any $\varepsilon >0$ and $F\in L^2(X,\nu _\varepsilon ;H)$, there exists a sequence $(F_n)_{n}\subseteq L^2(X,\nu _\varepsilon ;H)$ such that $\mathbf{T}_\varepsilon (t)F_n$ belongs to ${\mathcal {F}}C^3_b(X;H)$ and

  $$\begin{aligned}&\lim _{n\rightarrow \infty }{\left\| F_n-F\right\| }_{L^{2}(X,\nu _\varepsilon ;H)}=0,\nonumber \\&\lim _{n\rightarrow \infty }{\left\| \mathbf{T}_{\varepsilon }(t)F_n-\mathbf{T}_\varepsilon (t) F\right\| }_{D^{1,2}(X,\nu _\varepsilon ;H)}=0,\qquad \;\, t>0. \end{aligned}$$

  (12)

  If, in addition, $F\in D^{1,2}(X,\nu _\varepsilon ;H)$ then $D_H F_n$ converges to $D_H F$ in $L^1(X, \nu _\varepsilon ; {\mathcal {H}}_2)$, as $n\rightarrow \infty$.

• (ii) For any $F\in L^2(\varOmega ,\nu ;H)$ there exists an infinitesimal sequence $(\varepsilon _n)_{n\in {{\mathbb {N}}}}$ such that $\mathbf{T}_{\varepsilon _n}(t) {\widetilde{F}}$ weakly converges to $\mathbf{T}_\varOmega (t) F$ in $D^{1,2}(\varOmega ,\nu ;H)$, where ${\widetilde{F}}$ is any $L^2$-extension of F to X.

Proof

• (i) Let $F=\sum _{i=1}^{\infty }f^{(i)}e_i$ where $f^{(i)}\in L^2(X,\nu _\varepsilon )$, $i\in {{\mathbb {N}}}$. For every $i\in {{\mathbb {N}}}$, by Theorem 1(i), there exists $(f_k^{(i)})_{k\in {{\mathbb {N}}}}\subseteq L^2(X,\nu _\varepsilon )$ converging to $f^{(i)}$ in $L^2(X,\nu _\varepsilon )$ such that $T_\varepsilon (t)f_k^{(i)}$ belongs to ${\mathcal {F}}C^3_b(X)$ and

  $$\begin{aligned} \lim _{k\rightarrow \infty }\Vert T_\varepsilon (t)f_k^{(i)}-T_\varepsilon (t)f^{(i)}\Vert _{D^{1,2}(X,\nu _\varepsilon )}=0,\qquad t, \varepsilon >0. \end{aligned}$$

  (13)

  Observe that (12) follows immediately from (13). Now fix $i,n\in {{\mathbb {N}}}$ and consider $k_i\in {{\mathbb {N}}}$ such that for every $k\ge k_i$ it holds

  $$\begin{aligned} \int _X|f_k^{(i)}-f^{(i)}|^2\mathrm{d}\nu _\varepsilon <\frac{1}{n2^i}. \end{aligned}$$

  Consider the vector field $F_n:=\sum _{i=1}^n f_{k_i}^{(i)}e_i$. We claim that $(F_n)$ is the sequence we are looking for. Indeed $F_n$ belongs to $L^2(X,\nu _\varepsilon ;H)$ for any $n\in {{\mathbb {N}}}$. Let $n_0\in {{\mathbb {N}}}$ be such that $\sum _{i=n_0+1}^{\infty }\Vert f^{(i)}\Vert _{L^2(X,\nu _\varepsilon )}^2\le \eta /2$ and let $n\ge n_0$ such that $1/n<\eta /2$. We have

  $$\begin{aligned} {\left\| F_n-F\right\| }_{L^2(X,\nu _\varepsilon ;H)}^2&\le \sum _{i=1}^n\int _X|f_{k_i}^{(i)}-f^{(i)}|^2\mathrm{d}\nu _\varepsilon +\sum _{i=n+1}^{\infty }\Vert f^{(i)}\Vert _{L^2(X,\nu _\varepsilon )}^2\\&\le \frac{1}{n}+\frac{\eta }{2}\le \eta . \end{aligned}$$

  In a similar way we can prove the other statements.

• (ii) is an immediate consequence of Proposition 4 and Theorem 1(ii). $\square$

Before going on, recall that usually in the characterisation of functions of bounded variation in terms of the short-time behaviour of suitable semigroups a crucial tool is an appropriate commutation formula between the semigroup and the gradient operator. For instance, for the Wiener space and the Ornstein–Uhlenbeck semigroup the equality $D_H S(t)f=e^{-t}{\mathbf {S}}(t)D_H f$ holds true for any $t \ge 0$. Let us prove a (quasi) commutation formula between $T_{\varOmega }(t)$ and $D_H$, under the following additional assumption.

Hypotheses 4

The map $(d_{\varOmega })^{-2}\Vert D_H^2d_{\varOmega }^2\Vert _{{{\mathcal {H}}}_{2}}$ belongs to $L^2(X, \nu )$.

Remark 3

It is not difficult to show that every open ball and every open ellipsoid of X as well as every open hyperplane of X satisfy Hypothesis 4. We show that Hypothesis 4 is satisfied when ${\varOmega }$ is the unit ball $B_X$ centred at zero. The other examples can be discussed in a similar fashion. Observe that, by Proposition 2, $\Vert D_H^2d_{\varOmega }^2\Vert _{{\mathcal H}_{2}}\le 2$ and $\Vert D_H^2d_{\varOmega }^2(x)\Vert _{{{\mathcal {H}}}_{2}}=0$ if $x\in B_X$. Moreover, there exists a constant $C>0$ such that

$$\begin{aligned} d_{B_X}(x)&\ge C\inf \{|h|_X\,|\, h\in H\cap (B_X-x)\}\ge C\inf \{|h|_X\,|\, h\in (B_X-x)\}\\&= C\inf \{|x-h|_X\,|\, h\in B_X\}=C\mathrm{dist}(x,B_X)=C||x|_X-1|, \end{aligned}$$

where $\mathrm{dist}(x,B_X)$ is the distance of x from $B_X$. So

$$\begin{aligned}&\int _X d_{B_X}^{-4}\Vert D_H d^2_{B_X}\Vert ^2_{{\mathcal {H}}_2}\mathrm{d}\nu \le K\int _{X\smallsetminus B_X}\frac{1}{(|x|_X-1)^4}\mathrm{d}\nu (x) \nonumber \\&\quad \le \! K\!\!\int _{X}\!\frac{1}{(|x|_X-1)^4} e^{-U(x)}\mathrm{d}\gamma (x) \le K\Bigl (\int _X\! e^{-p'U}\mathrm{d}\gamma \Bigr )^{\frac{1}{p'}}\! \Bigl (\int _{X}\!\frac{1}{(|x|_X-1)^{4p}}\mathrm{d}\gamma (x)\Bigr )^{\frac{1}{p}}\nonumber \nonumber \\&\quad \le K\Bigl (\int _X e^{-p'U}\mathrm{d}\gamma \Bigr )^{\frac{1}{p'}} \Bigl (\int _{{{\mathbb {R}}}^n}\frac{1}{(|\xi |_{{{\mathbb {R}}}^n}-1)^{4p}}\mathrm{d}\gamma _n(\xi )\Bigr )^{\frac{1}{p}}, \end{aligned}$$

(14)

where K is a positive constant and $\gamma _n$ denotes the n-dimensional Gaussian measure, image of $\gamma$ under the projection on $\mathrm{span}\,\{v_1,\ldots ,v_n\}$. To conclude, observe that there exists $n\in {{\mathbb {N}}}$ such that the right-hand side of (14) is finite.

Proposition 5

Under Hypotheses 1, 2, 3 and 4, the formula

$$\begin{aligned}&D_H T_\varOmega (t)f -(e^{-tQ_\infty ^{-1}}{} \mathbf{T}_{\varOmega }(t)D_H f)\nonumber \\&\quad =-\int _0^te^{(s-t)Q_\infty ^{-1}}\mathbf{T}_\varOmega (t-s)(D_H^2 UD_H T_\varOmega (s)f)\mathrm{d}s. \end{aligned}$$

(15)

holds true $\nu$-a.e. in ${\varOmega }$, for any $f\in \mathrm{Lip}_c({\varOmega })$ and $t>0$.

Proof

In order to prove (15) we show that

$$\begin{aligned}&\int _{\varOmega }\langle D_H T_\varOmega (t)f ,G\rangle _H \mathrm{d}\nu = \int _{\varOmega }\langle e^{-tQ_\infty ^{-1}}{} \mathbf{T}_{\varOmega }(t)D_H f, G\rangle _H \mathrm{d}\nu \nonumber \\&\quad -\int _{\varOmega }\int _0^t\langle e^{(s-t)Q_\infty ^{-1}}(\mathbf{T}_\varOmega (t-s)(D_H^2 UD_H T_\varOmega (s)f)), G\rangle _H \mathrm{d}s \mathrm{d}\nu , \end{aligned}$$

(16)

for any $f\in \mathrm{Lip}_c({\varOmega })$, $G \in C_b({\varOmega };H)$ and $t>0$. By performing slight changes in [12, Appendix A] we get

$$\begin{aligned}&D_H T_\varepsilon (t)g -(e^{-tQ_\infty ^{-1}}{} \mathbf{T}_{\varepsilon }(t)D_H g)\nonumber \\&\quad =-\int _0^te^{(s-t)Q_\infty ^{-1}}\mathbf{T}_\varepsilon (t-s)(D_H^2 \varPhi _\varepsilon D_H T_\varepsilon (s)g)\mathrm{d}s \end{aligned}$$

(17)

$\nu _\varepsilon$-a.e. in X for any $g\in \mathrm{\mathrm{Lip}}_b(X)$ and $\varepsilon >0$, where $T_\varepsilon (t)$ is the semigroup introduced in Sect. 2.1. Now, let $f\in \mathrm{Lip}_c({\varOmega })$ and ${\widetilde{f}}$ be the trivial extension to zero of f in the whole space X. Clearly, ${\widetilde{f}}$ belongs to $\mathrm{\mathrm{Lip}}_b(X)$ and (17) holds true with g replaced by ${\widetilde{f}}$. Consequently, multiplying (17) by the function G and integrating on ${\varOmega }$ with respect to $\nu$ yield

$$\begin{aligned}&\int _{\varOmega }\langle D_H T_\varepsilon (t){\widetilde{f}}, G\rangle _H \mathrm{d}\nu =\int _{\varOmega }\langle e^{-tQ_\infty ^{-1}}(\mathbf{T}_{\varepsilon }(t)D_H {\widetilde{f}}) , G\rangle _H \mathrm{d}\nu \nonumber \\&\qquad -\int _{\varOmega }\int _0^t\langle e^{(s-t)Q_\infty ^{-1}}(\mathbf{T}_\varepsilon (t-s)(D_H^2 \varPhi _\varepsilon D_H T_\varepsilon (s){\widetilde{f}})), G\rangle _H \mathrm{d}s \mathrm{d}\nu \nonumber \\&\quad =\int _{\varOmega }\langle e^{-tQ_\infty ^{-1}}\mathbf{T}_{\varepsilon }(t)D_H {\widetilde{f}}, G\rangle _H \mathrm{d}\nu \nonumber \\&\qquad -\int _0^t\int _{\varOmega }\langle e^{(s-t)Q_\infty ^{-1}}(\mathbf{T}_\varepsilon (t-s)(D_H^2 \varPhi _\varepsilon D_H T_\varepsilon (s){\widetilde{f}})), G\rangle _H \mathrm{d}\nu \mathrm{d}s , \end{aligned}$$

(18)

where in the last line we used the Fubini–Tonelli theorem.

The proof of (16) is split in two steps.

Step 1. We argue by approximation on the last terms in (18) and (16).

For every $\varepsilon ,s>0$ we fix a Borel measurable version of $D_HT_{\varOmega }(s)f$ and $D_HT_\varepsilon (s){\widetilde{f}}$ in $L^2(\varOmega ,\nu ;H)$ and $L^2(X,\nu _\varepsilon ;H)$, respectively. Consider the function

$$\begin{aligned} \varGamma _\varepsilon (s,x):={\left\{ \begin{array}{ll}D_HT_{\varOmega }(s)f(x), &{}x\in {\varOmega };\\ D_HT_\varepsilon (s){\widetilde{f}}(x), &{} x\in X\smallsetminus {\varOmega }.\end{array}\right. } \end{aligned}$$

Observe that the map $x\mapsto \varGamma _\varepsilon (s,x)$ is an extension of $D_HT_{\varOmega }(s)f$ to the whole X. Thus, by Theorem 2 there is a sequence $\varepsilon _n\downarrow 0$ such that for every $\eta >0$ the function $\mathbf{T}_{\varepsilon _n}(t-s)(D_H^2 U\varGamma _\eta (s,\cdot ))$ weakly converges to $\mathbf{T}_{{\varOmega }}(t-s)(D_H^2 UD_HT_{\varOmega }(s)f)$ in $D^{1,2}(\varOmega ,\nu ;H)$. Observe that the set

$$\begin{aligned} \{\mathbf{T}_{\varepsilon _n}(t-s)(D_H^2 U\varGamma _\eta (s,\cdot ))\,|\, n\in {{\mathbb {N}}}\text { and }s,\eta >0\} \end{aligned}$$

(19)

is bounded in $L^2(\varOmega ,\nu ;H)$. Indeed by the contractivity of $\mathbf{T}_{\varepsilon _n}(t)$ in the space $L^2(X,\nu _{\varepsilon _n};H)$, the fact that $U\equiv \varPhi _\varepsilon$ on ${\varOmega }$ and estimate (8) we have

$$\begin{aligned}&{\left\| \mathbf{T}_{\varepsilon _n}(t-s)(D_H^2 U\varGamma _\eta (s,\cdot ))\right\| }_{L^2(\varOmega ,\nu ;H)}\\&\quad ={\left\| \mathbf{T}_{\varepsilon _n}(t-s)(D_H^2 U\varGamma _\eta (s,\cdot ))\right\| }_{L^2(\varOmega ,\nu _{\varepsilon _n};H)}\\&\quad \le {\left\| \mathbf{T}_{\varepsilon _n}(t-s)(D_H^2 U\varGamma _\eta (s,\cdot ))\right\| }_{L^2(X,\nu _{\varepsilon _n};H)}\\&\quad \le {\left\| D_H^2 U\varGamma _\eta (s,\cdot )\right\| }_{L^2(X,\nu _{\varepsilon _n};H)} \le [D_HU]_{H\text {-Lip}}{\left\| \varGamma _\eta (s,\cdot )\right\| }_{L^2(X,\nu _{\varepsilon _n};H)}\\&\quad \le [D_HU]_{H\text {-Lip}} \left( \Vert D_HT_{\eta }(s){\widetilde{f}}\Vert _{L^2(X,\nu _{\varepsilon _n};H)} +{\left\| D_HT_{\varOmega }(s)f\right\| }_{L^2({\varOmega },\nu ;H)}\right) \\&\quad \le [D_HU]_{H\text {-Lip}} e^{-2\lambda _1^{-1}s}{\left( \Vert T_{\eta }(s)|D_H{\widetilde{f}}|_H\Vert _{L^2(X,\nu _{\varepsilon _n})} +\Vert T_{\varOmega }(s)|D_H f|_H\Vert _{L^2(\varOmega ,\nu )}\right) }\\&\quad \le 2[D_HU]_{H\text {-Lip}}(\nu (X))^{\frac{1}{2}}\Vert D_H f\Vert _{L^\infty (\varOmega ,\nu ;H)} , \end{aligned}$$

where in the last line we used the contractivity of $T_\eta (t)$ and $T_{\varOmega }(t)$ in $L^\infty$ and the fact that $\nu _{\varepsilon _n}(X)\le \nu (X)$ for any $n\in {{\mathbb {N}}}$. So there exists $M>0$ large enough so that the family in (19) is contained in B(0, M), the ball of $L^2(\varOmega ,\nu ;H)$ with centre 0 and radius M.

Recall that every bounded subset of $L^2(\varOmega ,\nu ;H)$ is weakly metrisable (see [15, Proposition 3.106]) and let $\rho :B(0,M)\times B(0,M)\rightarrow {{\mathbb {R}}}$ be a metric such that the topology generated by $\rho$ and the weak topology in B(0, M) coincide. Now we use a diagonal argument to pass to the limit in (18). Let $n_1\in {{\mathbb {N}}}$ be such that for every $n\ge n_1$ it holds

$$\begin{aligned} \rho {\left( \mathbf{T}_{\varepsilon _n}(t-s)(D_H^2 U{\overline{\varGamma }}_1(s,\cdot )), \mathbf{T}_{{\varOmega }}(t-s)(D_H^2 UD_HT_{\varOmega }(s)f)\right) }\le 1 , \end{aligned}$$

where ${\overline{\varGamma }}_j(s,x)= \varGamma _{j^{-1}}(s,x)$ for any $s>0$ and $x\in X$. Now assume that $n_1,\ldots , n_k$ are already constructed and consider $n_{k+1}>n_k$such that for every $n\ge n_{k+1}$

$$\begin{aligned} \rho {\left( \mathbf{T}_{\varepsilon _n}(t-s)(D_H^2 U{\overline{\varGamma }}_{n_k}(s,\cdot )), \mathbf{T}_{{\varOmega }}(t-s)(D_H^2 UD_HT_{\varOmega }(s)f)\right) } \le \frac{1}{2^k}. \end{aligned}$$

Consider now the sequence $(\mathbf{T}_{\varepsilon _{n_k}}(t-s)(D_H^2 U{\overline{\varGamma }}_{n_k}(s,\cdot )))_{k\in {{\mathbb {N}}}}$ and observe that it weakly converges to $\mathbf{T}_{{\varOmega }}(t-s)(D_H^2 UD_HT_{\varOmega }(s)f)$ in $L^2(\varOmega ,\nu ;H)$ as $k\rightarrow \infty$.

Step 2. To complete the proof, we replace $\varepsilon$ in (18) by a sequence $\varepsilon _{m}\downarrow 0$ such that step 1 and Theorems 1, 2 apply. Let us show that we can take the limit as $m\rightarrow \infty$. Indeed, from Theorem 1 it follows that for any $f \in L^2({\varOmega }, \nu )$, $T_{\varepsilon _m}(t) {\widetilde{f}}$ weakly converges (up to a subsequence) to $T_\varOmega (t)f$ in $D^{2,2}(\varOmega ,\nu )$ as $m \rightarrow \infty$, hence writing $T_m, \mathbf{T}_m, \varPhi _m, \varGamma _m$ in place of $T_{\varepsilon _m}, \mathbf{T}_{\varepsilon _m}, \varPhi _{\varepsilon _m}, \varGamma _{\varepsilon _m}$ we obtain

$$\begin{aligned} \lim _{m \rightarrow \infty }\int _{\varOmega }\langle D_H T_{m}(t){\widetilde{f}}, G\rangle _H \mathrm{d}\nu = \int _{\varOmega }\langle D_H T_{\varOmega }(t)f, G\rangle _H \mathrm{d}\nu \end{aligned}$$

and by the analogous vector-valued result (see Theorem 2, (10) and again (7))

$$\begin{aligned} \lim _{m \rightarrow \infty } \int _{\varOmega }\langle e^{-tQ_\infty ^{-1}}\mathbf{T}_{m}(t)D_H {\widetilde{f}}, G\rangle _H \mathrm{d}\nu = \int _{\varOmega }\langle e^{-tQ_\infty ^{-1}}{} \mathbf{T}_{\varOmega }(t)D_H f, G\rangle _H \mathrm{d}\nu . \end{aligned}$$

To conclude we have to prove that the last term in the right-hand side of (18) converges to the last term in the right-hand side of (16).

$$\begin{aligned}&{\left| \int _{\varOmega }\!\langle {\mathbf {T}}_m(t-s)(D^2_H\varPhi _m D_HT_m(s){\widetilde{f}})\mathrm{d}\nu \!-\!\!\int _{\varOmega }\!\!\mathbf{T}_\varOmega (t-s)(D_H^2 U D_H T_\varOmega (s)f),G\rangle _H \mathrm{d}\nu \right| }\\&\quad \le {\left| \int _{\varOmega }\langle {\mathbf {T}}_m(t-s)(D^2_H\varPhi _m D_HT_m(s){\widetilde{f}}) -\mathbf{T}_m(t-s)(D_H^2 \varPhi _m \varGamma _m(s,\cdot )),G\rangle _H\mathrm{d}\nu \right| }\\&\qquad + {\left| \int _{\varOmega }\langle \mathbf{T}_m(t-s)(D_H^2 \varPhi _m \varGamma _m(s,\cdot )) -\mathbf{T}_m(t-s)(D_H^2 U \varGamma _m(s,\cdot )),G\rangle _H\mathrm{d}\nu \right| }\\&\qquad + {\left| \int _{\varOmega }\langle \mathbf{T}_m(t-s)(D_H^2 U\varGamma _m(s,\cdot )) -\mathbf{T}_\varOmega (t-s)(D_H^2 U D_H T_\varOmega (s)f),G\rangle _H\mathrm{d}\nu \right| }\\&\quad =:I_1(m)+I_2(m)+I_3(m) . \end{aligned}$$

Let us estimate $I_1$. Using that $\varPhi _m\equiv U$ on ${\varOmega }$ for every $m\in {{\mathbb {N}}}$, formula (10) and the invariance property of $T_m$ with respect to $\nu _m:=\nu _{\varepsilon _m}$ we have

$$\begin{aligned}&I_1(m) \le \int _{\varOmega }\Big |{\mathbf {T}}_m(t-s)(D^2_H\varPhi _m D_HT_m(s){\widetilde{f}}) \nonumber \\&\qquad \qquad -\mathbf{T}_m(t-s)(D_H^2 \varPhi _m \varGamma _m(s,\cdot ))\Big |_H{\left| G\right| }_H\mathrm{d}\nu \nonumber \\&\quad \le (\nu (X))^{\frac{1}{2}}{\left\| G\right\| }_\infty \! {\left( \!\int _{\varOmega }{\left| {\mathbf {T}}_m(t-s)(D^2_H\varPhi _m D_HT_m(s){\widetilde{f}}-D_H^2 \varPhi _m \varGamma _m(s,\cdot ))\right| }^2_H\mathrm{d}\nu \!\right) }^{\frac{1}{2}}\nonumber \\&\quad \le (\nu (X))^{\frac{1}{2}}{\left\| G\right\| }_\infty \! {\left( \!\int _{\varOmega }T_m(t-s){\left| D^2_H\varPhi _m D_HT_m(s){\widetilde{f}}-D_H^2 \varPhi _m \varGamma _m(s,\cdot )\right| }^2_H\mathrm{d}\nu _m\!\right) }^{\frac{1}{2}} \nonumber \\&\quad \le (\nu (X))^{\frac{1}{2}}{\left\| G\right\| }_\infty \! {\left( \int _X T_m(t-s){\left| D^2_H\varPhi _m D_HT_m(s){\widetilde{f}}-D_H^2 \varPhi _m \varGamma _m(s,\cdot )\right| }^2_H\mathrm{d}\nu _m\!\right) }^{\frac{1}{2}} \nonumber \\&\quad = (\nu (X))^{\frac{1}{2}}{\left\| G\right\| }_\infty {\left( \int _X {\left| (D^2_H\varPhi _m D_HT_m(s){\widetilde{f}})-(D_H^2 \varPhi _m \varGamma _m(s,\cdot ))\right| }^2_H\mathrm{d}\nu _m\right) }^{\frac{1}{2}} \nonumber \\&\quad \le (\nu (X))^{\frac{1}{2}}{\left\| G\right\| }_\infty {\left( \int _\varOmega \Vert D_H^2 U\Vert _{{\mathcal {H}}_2}^2 {\left| D_H T_m(s){\widetilde{f}} -D_H T_\varOmega (s)f\right| }^2_H\mathrm{d}\nu \right) }^{\frac{1}{2}} \nonumber \\&\quad \le [D_H U]_{H\text {-Lip}}(\nu (X))^{\frac{1}{2}}{\left\| G\right\| }_\infty {\left( \int _\varOmega {\left| D_H T_m(s){\widetilde{f}}-D_H T_\varOmega (s)f\right| }^2_H\mathrm{d}\nu \right) }^{\frac{1}{2}}. \end{aligned}$$

(20)

The right-hand side of (20) converges to zero as $m\rightarrow \infty$: indeed, $D_H T_m(s){\widetilde{f}}$ converges pointwise $\nu$-almost everywhere in $\varOmega$ to $D_H T_\varOmega (s)f$. Furthermore, by Proposition 3 we have that $\nu$-a.e. in $\varOmega$

$$\begin{aligned}&|D_H T_{\varepsilon _m}(s){\widetilde{f}}-D_HT_\varOmega (s)f|^2_H \le 2\left( |D_H T_{\varepsilon _m}(s){\widetilde{f}}|^2_H+|D_HT_\varOmega (s)f|^2_H\right) \\&\quad \le 2e^{-2\lambda _1^{-1}s}{\left( T_{\varepsilon _m}(s)|D_H {\widetilde{f}}|_H^2+T_{{\varOmega }}(s)|D_H f|_H^2\right) } \le 2{\left\| D_H f\right\| }_{L^\infty (\varOmega ,\nu ;H)}. \end{aligned}$$

So by the dominated convergence theorem we get that $I_1(m)$ vanishes as $m\rightarrow \infty$. Now, using similar arguments we can estimate $I_2(m)$ as follows

$$\begin{aligned} I_2(m)&\le (\nu (X))^{\frac{1}{2}}{\left\| G\right\| }_\infty \Bigl (\int _X {\left| (D_H^2 \varPhi _m \varGamma _m(s,\cdot ))-(D_H^2 U \varGamma _m(s,\cdot ))\right| }^2_H\mathrm{d}\nu _m\Bigr )^{\frac{1}{2}} \nonumber \\&\le (\nu (X))^{\frac{1}{2}}{\left\| G\right\| }_\infty \Bigl (\int _X \Vert D_H^2 \varPhi _m-D_H^2U\Vert _{{\mathcal {H}}_2}^2 |\varGamma _m(s,\cdot )|_H^2\mathrm{d}\nu _m\Bigr )^{\frac{1}{2}} \nonumber \\&\le (\nu (X))^{\frac{1}{2}}{\left\| G\right\| }_\infty \Bigl (\int _X \Vert D_H^2 \varPhi _m-D_H^2U\Vert _{{\mathcal {H}}_2}^2 |(D_HT_m(s){\widetilde{f}})\chi _{X\smallsetminus \varOmega } \nonumber \\&\quad +(D_HT_\varOmega (s)f)\chi _\varOmega |_H^2\mathrm{d}\nu _m\Bigr )^{\frac{1}{2}} \nonumber \\&\le (2\nu (X))^{\frac{1}{2}}{\left\| G\right\| }_\infty \Bigl (\int _X \Vert D_H^2 \varPhi _m-D_H^2U\Vert _{{\mathcal {H}}_2}^2 \bigl (|D_HT_m(s){\widetilde{f}}|^2_H \nonumber \\&\quad +|D_HT_\varOmega (s)f|_H^2\chi _\varOmega \bigr )\mathrm{d}\nu _m\Bigr )^{\frac{1}{2}} \nonumber \\&\le (2\nu (X))^{\frac{1}{2}}{\left\| G\right\| }_\infty \Bigl (\int _X \Vert D_H^2 \varPhi _m-D_H^2U\Vert _{{\mathcal {H}}_2}^2 \bigl (T_m(s)|D_H{\widetilde{f}}|^2_H \nonumber \\&\quad +(T_\varOmega (s)|D_Hf|_H^2)\chi _\varOmega \bigr )\mathrm{d}\nu _m\Bigr )^{\frac{1}{2}} \nonumber \\&\le 2(\nu (X))^{\frac{1}{2}}{\left\| D_Hf\right\| }_{L^\infty (\varOmega ,\nu ;H)}{\left\| G\right\| }_\infty \Bigl (\int _X \Vert D_H^2 \varPhi _m-D_H^2U\Vert _{{\mathcal {H}}_2}^2 \mathrm{d}\nu _m\Bigr )^{\frac{1}{2}} \nonumber \\&= 2(\nu (X))^{\frac{1}{2}}{\left\| D_Hf\right\| }_{L^\infty (\varOmega ,\nu ;H)}{\left\| G\right\| }_\infty \! \Bigl (\!\int _X\!\! \frac{1}{4\varepsilon _m} \Vert D_H^2 d_\varOmega ^2\Vert _{{\mathcal {H}}_2}^2 e^{-U-\frac{1}{2\varepsilon _m}d^2_\varOmega }\mathrm{d}\gamma \!\Bigr )^{\frac{1}{2}}\!\!. \end{aligned}$$

(21)

Now observe that the right-hand side of (21) vanishes as $m\rightarrow \infty$. Indeed the function $\frac{1}{4\varepsilon _m}\Vert D_H^2 d_\varOmega ^2\Vert _{{\mathcal {H}}_2}^2 e^{-U-\frac{1}{2\varepsilon _m}d^2_\varOmega }$ identically vanishes in ${\varOmega }$ and converges pointwise to 0 $\nu$-almost everywhere in $X{\setminus } \varOmega$ as $m\rightarrow \infty$. Furthermore, observe that the function $(0,\infty )\ni \varepsilon \mapsto R(\varepsilon ):=\frac{1}{4\varepsilon ^2}{\left| D^2_Hd_\varOmega ^2\right| }^2e^{-\frac{1}{2{\varepsilon }}d_\varOmega ^2}$ attains its maximum in $\varepsilon =d_\varOmega ^2/4$ where it equals to $4d_\varOmega ^{-4}\Vert D_H^2 d_\varOmega ^2|_{{\mathcal {H}}_2}^2$. Thus, using Hypothesis 4 and applying the dominated convergence theorem we infer that also $I_2(m)$ converges to zero as m goes to infinity.

Finally $I_3(m)$ converges to zero as m goes to infinity thanks to step 1 and this concludes the proof. $\square$

Corollary 1

Assume Hypotheses 1, 2, 3 and 4 hold true. For any $t>0$ and $p>1$ there exist two operators $\mathbf{S}_1(t): L^p(\varOmega ,\nu ;H)\rightarrow L^1(\varOmega ,\nu ;H)$ and $S_2(t): L^p(\varOmega ,\nu )\rightarrow L^1(\varOmega ,\nu ;H)$ such that for every continuous and H-differentiable function $\varphi :\varOmega \rightarrow {{\mathbb {R}}}$ with H-Lipschitz gradient

$$\begin{aligned} D_HT_\varOmega (t)\varphi =\mathbf{S}_1(t)D_H\varphi +S_2(t)\varphi . \end{aligned}$$

Moreover, the adjoint operator $(\mathbf{S}_1(t))^*$ maps $\mathrm{\mathrm{Lip}}_c(\varOmega ;H)$ into $L^{\infty }(\varOmega ,\nu ;H)$ and verifies ${\left\| \mathbf{S}_1^*(t)F\right\| }_\infty \le C_1(t){\left\| F\right\| }_\infty$ for any $F\in \mathrm{\mathrm{Lip}}_c(\varOmega ;H)$ with $C_1(t)\rightarrow 1$ as $t\rightarrow 0$ and the norm $C_2(t):={\left\| S_2(t)\right\| }_{{\mathcal {L}}(L^p,L^1)}\rightarrow 0$ as $t\rightarrow 0$.

Proof

Setting $\mathbf{S}_1(t):=e^{-tQ_\infty ^{-1}}{} \mathbf{T}_\varOmega (t)$, [6, Proposition 1.10] yields that $\mathbf{S}_1^*(t)=\mathbf{T}_\varOmega (t)e^{-tQ_\infty ^{-1}}$ maps $\mathrm{Lip}_c(\varOmega ;H)$ into $L^{\infty }(\varOmega ,\nu ;H)$ and

$$\begin{aligned} {\left\| \mathbf{S}_1^*(t)F\right\| }_\infty&=\Vert (e^{-tQ_\infty ^{-1}}{} \mathbf{T}_\varOmega (t))^*F\Vert _\infty \nonumber \\&=\Vert \mathbf{T}_\varOmega (t)e^{-tQ_\infty ^{-1}}F\Vert _\infty \le |e^{-tQ_\infty ^{-1}}|_{{{\mathcal {L}}}(H)}{\left\| F\right\| }_\infty . \end{aligned}$$

(22)

Moreover, setting $S_2(t):=-\int _0^te^{(s-t)Q_\infty ^{-1}}\mathbf{T}_{\varOmega }(t-s)(D_H^2 U D_H T_{\varOmega }))\mathrm{d}s$, by the contractivity of ${\mathbf {T}}_\varOmega (t)$ in $L^1(\varOmega ,\nu ; H)$, (7), Hypothesis 2, the contractivity of $e^{-tQ^{-1}_\infty }$ in H, estimate (11) and the invariance property of $T_\varOmega (t)$, we get

$$\begin{aligned}&{\left\| \mathbf {S}_2(t) \varphi \right\| }_{L^1(\varOmega ,\nu ;H)} \nonumber \\ {}&\quad \le \int _0^t\int _\varOmega {\left| e^{(s-t)Q_\infty ^{-1}}{} \mathbf {T}_\varOmega (t-s)(D^2_HUD_HT_\varOmega (s)\varphi )\right| }_H\mathrm {d}\nu \mathrm {d}s \nonumber \\ {}&\quad \le \int _0^t\left| e^{(s-t)Q_\infty ^{-1}}\right| _{{{\mathcal {L}}}(H)} \int _\varOmega {\left| \mathbf {T}_\varOmega (t-s)(D^2_HUD_HT_\varOmega (s)\varphi )\right| }_H\mathrm {d}\nu \mathrm {d}s \nonumber \\ {}&\quad \le \int _0^t \int _\varOmega {\left| \mathbf {T}_\varOmega (t-s)(D^2_HUD_HT_\varOmega (s)\varphi )\right| }_H\mathrm {d}\nu \mathrm {d}s \nonumber \\ {}&\quad \le \int _0^t \int _\varOmega {\left| D^2_HUD_HT_\varOmega (s)\varphi \right| }_H\mathrm {d}\nu \mathrm {d}s \nonumber \\ {}&\quad \le \int _0^t \int _\varOmega \Vert D^2_HU\Vert _{{\mathcal {H}}_2}{\left| D_HT_\varOmega (s)\varphi \right| }_H\mathrm {d}\nu \mathrm {d}s \nonumber \\ {}&\quad \le \int _0^t {\left( \int _\varOmega \Vert D^2_HU\Vert ^{p'}_{{\mathcal {H}}_2}\mathrm {d}\nu \right) }^{\frac{1}{p'}} {\left( \int _\varOmega {\left| D_HT_\varOmega (s)\varphi \right| }^p_H\mathrm {d}\nu \right) }^{\frac{1}{p}} \mathrm {d}s \nonumber \\ {}&\quad \le K_p^{\frac{1}{p}}\Vert D^2_HU\Vert _{L^{p'}(X,\nu ;{\mathcal {H}}_2)} \int _0^t s^{-\frac{1}{2}}{\left( \int _\varOmega T_\varOmega (s)|\varphi |^p\mathrm {d}\nu \right) }^{\frac{1}{p}} \mathrm {d}s \nonumber \\ {}&\quad \le K^{\frac{1}{p}}_p\Vert D^2_HU\Vert _{L^{p'}(X,\nu ;{\mathcal {H}}_2)}\Vert \varphi \Vert _{L^p(\varOmega ,\nu )} \int _0^t s^{-\frac{1}{2}} \mathrm {d}s \nonumber \\ {}&\quad = 2K^{\frac{1}{p}}_p\sqrt{t}\Vert D^2_HU\Vert _{L^{p'}(X,\nu ;{\mathcal {H}}_2)} \Vert \varphi \Vert _{L^p(\varOmega ,\nu )} \end{aligned}$$

(23)

for any $t>0$. By the assumption on U we deduce that the operator $S_2(t)$ is bounded from $L^p(\varOmega ,\nu )$ into $L^1(\varOmega ,\nu ;H)$ for any $t>0$. Finally, estimates (22) and (23) allow us to complete the proof. $\square$

Now, we are able to prove the main result of this section.

Theorem 3

Assume Hypotheses 1, 2, 3 and 4 hold true and let $u\in L^2({\varOmega },\nu )$. The following statements are true:

• (i) if $\liminf _{t \rightarrow 0^+}\Vert D_H T_{\varOmega }(t)u \Vert _{L^1({\varOmega },\nu ;H)}$ is finite, then $u\in BV(\varOmega ,\nu )$;

• (ii) if $u\in BV(\varOmega ,\nu )$, then $\limsup _{t \rightarrow 0^+}\Vert D_H T_{\varOmega }(t)u \Vert _{L^1({\varOmega },\nu ;H)}\le |D_\nu u|(\varOmega )$.

Hence, $u\in BV(\varOmega ,\nu )$ iff $\displaystyle \lim _{t\rightarrow 0^+}\Vert D_H T_{\varOmega }(t)u \Vert _{L^1({\varOmega },\nu ;H)}<\infty$. In this case

$$\begin{aligned} |D_\nu u|(\varOmega )=\lim _{t\rightarrow 0^+}\Vert D_H T_{\varOmega }(t)u \Vert _{L^1({\varOmega },\nu ;H)}. \end{aligned}$$

(24)

Proof

(i) follows from the strong continuity of $T_{{\varOmega }}(t)$ in $L^1({\varOmega }, \nu )$, see Proposition 3(i), and the lower semicontinuity of the norm (9), which imply

$$\begin{aligned} |D_\nu u|(\varOmega )\le \liminf _{t\rightarrow 0^+}\int _\varOmega |D_H T_\varOmega (t)u|_H\mathrm{d}\nu . \end{aligned}$$

To prove (ii) we write the $L^1$-norm of the gradient of $T_{\varOmega }(t)u$ by duality, as

Taking into account that, for any $F\in \mathrm{Lip}_b({\varOmega };H)$ we get

$$\begin{aligned}&\int _{\varOmega }\langle D_H T_\varOmega (t)u,F \rangle _H \mathrm{d}\nu = \int _{\varOmega }u (D_H T_\varOmega (t))^*F \mathrm{d}\nu \\&\quad \le \int _{\varOmega }u \Big (\mathbf{S}_1(t)D_H+S_2(t)\Big )^*F \mathrm{d}\nu = \int _{\varOmega }u (D_H^*\mathbf{S}_1(t)^*F +S_2(t)^*F) \mathrm{d}\nu \\&\quad \le |D_\nu u|(\varOmega )\Vert \mathbf{S}_1(t)^*F\Vert _{\infty }+ \Vert u\Vert _{L^2({\varOmega },\nu )}\Vert S_2(t)^*F\Vert _{L^2({\varOmega },\nu )}\\&\quad \le (|D_\nu u|(\varOmega )C_1(t)+C_2(t))\Vert F\Vert _\infty \end{aligned}$$

we deduce that

$$\begin{aligned} \Vert D_H T_\varOmega (t)u\Vert _{L^1({\varOmega }, \nu ;H)}\le C_1(t)|D_\nu u|(\varOmega )+C_2(t) \end{aligned}$$

(25)

for any $t>0$ where $C_i$ ($i=1,2$) are the positive functions in Corollary 1. Thus, taking the limsup as $t\rightarrow 0^+$ in (25) we get

$$\begin{aligned} \limsup _{t\rightarrow 0^+}\int _\varOmega |D_H T_\varOmega (t)u|_H\mathrm{d}\nu \le |D_\nu u|(\varOmega ) \end{aligned}$$

and the proof is complete. $\square$

It follows from Theorem 3 that functions in $BV({\varOmega },\nu )$ may be approximated in variation by smooth functions. This result was already known in infinite dimension when ${\varOmega }=X$ and $T_{\varOmega }(t)$ is the Ornstein–Uhlenbeck semigroup and in a convex set, see [21], where the approximation is based on finite-dimensional reductions of the semigroup generated by the Neumann Ornstein–Uhlenbeck operator in ${\varOmega }$.

Proposition 6

Under Hypotheses 1, 2, 3 and 4, for any $f\in BV({\varOmega },\nu )$ there exists a sequence $(f_n)_{n\in {{\mathbb {N}}}}\subseteq D^{1,2}({\varOmega },\nu )$ such that

$$\begin{aligned} \mathrm{(i)}\,\lim _{n\rightarrow \infty }\Vert f_n-f\Vert _{L^2({\varOmega },\nu )}=0\ \ \mathrm{and}\ \ \mathrm{(ii)}\,\lim _{n\rightarrow \infty }\int _{{\varOmega }}|D_H f_n|_H\mathrm{d}\nu =|D_\nu f|({\varOmega }).\nonumber \\ \end{aligned}$$

(26)

If $C\subseteq {\varOmega }$ is closed and $|D_\nu f|(\partial C)=0$ then $\displaystyle |D_\nu f|(C)=\lim _{n\rightarrow \infty }\int _{C}|D_H f_n|_H\mathrm{d}\nu .$

Proof

Consider the semigroup $T_{\varOmega }(t)$ generated in $L^2({\varOmega },\nu )$ by the operator $L_{\varOmega }$ defined in (4). It is known that for any $f\in L^2({\varOmega }, \nu )$ the function $T_{\varOmega }(t)f$ belongs to $D^{1,2}({\varOmega },\nu )$ for any $t>0$ and by the strong continuity of $T_{\varOmega }(t)$, $T_{\varOmega }(t)f$ converges to f in $L^2({\varOmega },\nu )$ as $t\rightarrow 0^+$. Moreover, Theorem 3 implies that $\Vert D_H T_{\varOmega }(t)f\Vert _{L^1({\varOmega },\nu ;H)}$ converges to $|D_\nu f|({\varOmega })$ as $t \rightarrow 0^+$. Thus, (26) is proved. To complete the proof let us observe that, by the lower semicontinuity of the total variation, for any open set $A\subseteq {\varOmega }$

$$\begin{aligned} |D_\nu f|(A)\le \liminf _{n\rightarrow \infty }\int _{A}|D_H f_n|_H\mathrm{d}\nu \end{aligned}$$

(27)

(see [21, Corollary 2.5]). Analogously we deduce that

$$\begin{aligned} |D_\nu f|(C)\ge \limsup _{n\rightarrow \infty }\int _{C}|D_H f_n|_H\mathrm{d}\nu \end{aligned}$$

(28)

for any closed subset $C\subseteq {\varOmega }$. Indeed, by (27) we obtain

$$\begin{aligned}&|D_\nu f|({\varOmega })-|D_\nu f|(C)=|D_\nu f|({\varOmega }{\setminus } C) \le \liminf _{n\rightarrow \infty }\int _{{\varOmega }{\setminus } C}|D_H f_n|_H\mathrm{d}\nu \\&\quad = \liminf _{n\rightarrow \infty }\left( \int _{\varOmega }|D_H f_n|_H\mathrm{d}\nu -\int _C|D_H f_n|_H\mathrm{d}\nu \right) \\&\quad = \lim _{n\rightarrow \infty }\int _{{\varOmega }}|D_H f_n|_H\mathrm{d}\nu -\limsup _{n \rightarrow \infty }\int _C|D_H f_n|_H\mathrm{d}\nu \end{aligned}$$

whence, using (26)(ii), estimate (28) follows. Now, using estimates (27), (28) and the fact that $|D_\nu f|(\partial C)=0$ we obtain

$$\begin{aligned} |D_\nu f|(C)=|D_\nu f|(\mathring{C})&\le \liminf _{n\rightarrow \infty }\int _{\mathring{C}}|D_H f_n|_H\mathrm{d}\nu \nonumber \\&\le \limsup _{n\rightarrow \infty }\int _{\mathring{C}}|D_H f_n|_H\mathrm{d}\nu \le |D_\nu f|(C), \end{aligned}$$

(29)

where $\mathring{C}$ denotes the interior of C. Estimate (29) yields the claim. $\square$

We conclude this section showing that estimate (8) and the previous approximation result allow to improve estimate (25) obtaining (30).

Theorem 4

Under Hypotheses 1, 2, 3 and 4, if $f\in BV({\varOmega }, \nu )$ then

$$\begin{aligned} \int _{\varOmega }|D_H T_{\varOmega }(t)f|_H \mathrm{d}\nu \le e^{-\lambda _1^{-1} t}|D_\nu f|({\varOmega }),\qquad \;\, t>0, \end{aligned}$$

(30)

$\lambda _1$ being the maximum eigenvalue of the covariance operator $Q_\infty$, see (2). Moreover, for any open set $A\subset {\varOmega }$ with ${\overline{A}}\subset \varOmega$,

$$\begin{aligned} \lim _{t \rightarrow 0^+}\int _A |D_H T_{\varOmega }(t)f|_H \mathrm{d}\nu =|D_\nu f|(A). \end{aligned}$$

Proof

Let $f\in BV({\varOmega },\nu )$ and let $(f_n)_{n\in {{\mathbb {N}}}}\in D^{1,2}({\varOmega },\nu )$ be the sequence given by Proposition 6. By the contractivity of $T_{\varOmega }(t)$ we deduce that $T_{\varOmega }(t)f_n$ converges to $T_{\varOmega }(t)f$ in $L^2({\varOmega }, \nu )$ as $n\rightarrow \infty$. This fact, together with the lower semicontinuity of the total variation, (8) and (6) yield

$$\begin{aligned}&\int _{{\varOmega }}|D_H T_{\varOmega }(t)f|_H\mathrm{d}\nu \le \liminf _{n\rightarrow \infty }\int _{{\varOmega }}|D_H T_{\varOmega }(t)f_n|_H\mathrm{d}\nu \\&\quad \le e^{-\lambda _1^{-1} t}\liminf _{n\rightarrow \infty }\int _{\varOmega }T_\varOmega (t)|D_H f_n|_H \mathrm{d}\nu \le e^{-\lambda _1^{-1} t}\lim _{n\rightarrow \infty }\int _{\varOmega }|D_H f_n|_H \mathrm{d}\nu \\&\quad = e^{-\lambda _1^{-1} t}|D_\nu f|({\varOmega }) \end{aligned}$$

whence (30) is proved. The last assertion follows immediately from Proposition 6 taking into account that $T_{\varOmega }(t)f$ satisfies (26). $\square$

4 Sets of finite perimeter in ${\varOmega }$

This section is devoted to provide some sufficient and necessary conditions in order that a Borel set $E \subseteq X$ have finite perimeter in ${\varOmega }$. We consider also the case of $BV({\varOmega },\nu )$ functions and ${\varOmega }=X$. There are three semigroups involved: beside $T_\varOmega (t)$, we consider the Ornstein–Uhlenbeck semigroup S(t) generated in $L^2(X, \gamma )$ by the realisation of the operator

$$\begin{aligned} L_{OU}\varphi = \mathrm{Tr}(D^2_H \varphi )-\sum _{i=1}^{\infty }\lambda _i^{-1}\langle x,e_i\rangle D_i\varphi \quad \varphi \in {\mathcal {F}}C^2_b(X) \end{aligned}$$

and the semigroup T(t) generated in $L^2(X,\nu )$ by the realisation of the operator

$$\begin{aligned} L\varphi = L_{OU}\varphi -\langle D_HU,D_H\varphi \rangle _H , \quad \varphi \in {\mathcal {F}}C^2_b(X) . \end{aligned}$$

(31)

Recall that S(t) admits a pointwise representation by means of the Mehler formula (1).

Theorem 5

Assume Hypotheses 1, 2, 3 and 4 hold true and let $E\subseteq X$ be a Borel set such that $P_\nu (E,{\varOmega })<\infty$. Then

$$\begin{aligned} \limsup _{t \rightarrow 0^+}\frac{1}{\sqrt{t}}\Vert T_{\varOmega }(t)\chi _E-\chi _E\Vert _{L^1({\varOmega }, \nu )}<\infty . \end{aligned}$$

(32)

More precisely

$$\begin{aligned} \Vert T_{\varOmega }(t)u-u\Vert _{L^1({\varOmega }, \nu )}\le 2 \sqrt{K_2t}|D_\nu u|(\varOmega ) \end{aligned}$$

(33)

for any $u \in BV({\varOmega }, \nu )$ and $t>0$ where $K_2$ is the constant in (7).

Proof

Clearly, once estimate (33) is proved, (32) follows at once choosing $u=\chi _E$. Thus, let us prove (33). To this aim, we consider $g\in L^\infty ({\varOmega }, \nu )$ and assume first that $u \in D^{1,2}({\varOmega },\nu )$. By the self-adjointness of the operators $L_{\varOmega }T_{\varOmega }(s)$ for $s\ge 0$ in $L^2({\varOmega },\nu )$ we have

$$\begin{aligned}&\int _{\varOmega }g(T_{\varOmega }(t)u-u)\mathrm{d}\nu = \int _{\varOmega }g\int _0^t \frac{\mathrm{d}}{\mathrm{d}s}T_{\varOmega }(s)u \mathrm{d}s \mathrm{d}\nu = \int _0^t \int _{\varOmega }g (L_{\varOmega }T_{\varOmega }(s)u) \mathrm{d}\nu \mathrm{d}s \nonumber \\&\quad = \int _0^t \int _{\varOmega }(L_{\varOmega }T_{\varOmega }(s)g) u \mathrm{d}\nu \mathrm{d}s = -\int _0^t \int _{\varOmega }\langle D_H T_{\varOmega }(s)g, D_H u\rangle _H \mathrm{d}\nu \mathrm{d}s. \end{aligned}$$

The Cauchy–Schwarz inequality and (7) yield

$$\begin{aligned} \int _{\varOmega }g(T_{\varOmega }(t)u-u)\mathrm{d}\nu&\le \int _0^t \int _{\varOmega }|D_H T_{\varOmega }(s)g|_H |D_H u|_H \mathrm{d}\nu \mathrm{d}s\\&=\int _0^t \int _{\varOmega }(|D_H T_{\varOmega }(s)g|^2_H)^{\frac{1}{2}} |D_H u|_H \mathrm{d}\nu \mathrm{d}s\\&\le \sqrt{K_2}\int _0^t s^{-\frac{1}{2}}\int _{\varOmega }(T_{\varOmega }(s)|g|^2)^{\frac{1}{2}} |D_H u|_H \mathrm{d}\nu \mathrm{d}s. \end{aligned}$$

From the contractivity of $T_{\varOmega }(t)$ in $L^\infty (\varOmega ,\nu )$, for any $t>0$ we deduce

$$\begin{aligned} \int _{\varOmega }g(T_{\varOmega }(t)u-u)\mathrm{d}\nu&\le \sqrt{K_2} {\left\| g\right\| }_{L^\infty ({\varOmega },\nu )} \int _0^t s^{-\frac{1}{2}}\int _{\varOmega }|D_H u|_H \mathrm{d}\nu \mathrm{d}s\nonumber \\&=2\sqrt{K_2 t}{\left\| g\right\| }_{L^\infty ({\varOmega },\nu )}\int _{\varOmega }|D_H u|_H \mathrm{d}\nu . \end{aligned}$$

(34)

For $u\in BV({\varOmega }, \nu )$, from Proposition 6 we get a sequence $u_n\in D^{1,2}({\varOmega }, \nu )$ converging to u in $L^2({\varOmega }, \nu )$ with $\lim _{n \rightarrow \infty }\int _{\varOmega }|Du_n|_H \mathrm{d}\nu =|D_\nu u|({\varOmega })$. Thus, putting $u_n$ in place of u in (34) and letting $n \rightarrow \infty$ we get

$$\begin{aligned} \int _{\varOmega }g(T_{\varOmega }(t)u-u)\mathrm{d}\nu&\le 2\sqrt{K_2t}{\left\| g\right\| }_{L^\infty ({\varOmega },\nu )} |D_\nu u|({\varOmega }), \qquad \;\,g\in L^\infty ({\varOmega },\nu ). \end{aligned}$$

Finally, taking the supremum on the $g\in L^\infty (\varOmega ,\nu )$ with $\Vert g\Vert _\infty \le 1$ we obtain

$$\begin{aligned} \int _{\varOmega }|T_{\varOmega }(t)u-u|\mathrm{d}\nu&\le 2\sqrt{K_2t} |D_\nu u|({\varOmega }) \end{aligned}$$

whence (33) follows. $\square$

Remark 4

Note that condition (32) is equivalent to

$$\begin{aligned} \limsup _{t \rightarrow 0^+}\frac{1}{\sqrt{t}}\int _{E^c\cap {\varOmega }} (T_{\varOmega }(t)\chi _E)\mathrm{d}\nu <\infty . \end{aligned}$$

Indeed, $|T_{\varOmega }(t)\chi _E-\chi _E|= (\chi _E-T_{\varOmega }(t)\chi _E)\chi _E+ (T_{\varOmega }(t)\chi _E-\chi _E)\chi _{E^c}$. The invariance of $T_{\varOmega }(t)$ with respect to $\nu$ in $\varOmega$ yields

$$\begin{aligned} \int _{\varOmega }(\chi _E-T_{\varOmega }(t)\chi _E)\chi _E\mathrm{d}\nu= & {} \int _{\varOmega }(T_{\varOmega }(t)\chi _E- \chi _ET_{\varOmega }(t)\chi _E)\mathrm{d}\nu \\= & {} \int _{\varOmega }\chi _{E^c}T_{\varOmega }(t)\chi _E \mathrm{d}\nu . \end{aligned}$$

Consequently,

$$\begin{aligned} \int _{\varOmega }|T_{\varOmega }(t)\chi _E-\chi _E|\mathrm{d}\nu = 2 \int _{{\varOmega }\cap E^c}T_{\varOmega }(t)\chi _E \mathrm{d}\nu . \end{aligned}$$

Now, we prove a quasi-converse of Theorem 5. We start with a preliminary result for bounded functions.

Proposition 7

Under Hypotheses 1, 2, 3 and 4, let $u\in L^\infty (X,\nu )$ be such that

$$\begin{aligned} \liminf _{t \rightarrow 0^+}\frac{1}{\sqrt{t}}\int _{\varOmega }\int _X|u(e^{-t}x+\sqrt{1-e^{-2t}y})-u(x)|\mathrm{d}\gamma (y)\mathrm{d}\nu (x) = C < \infty . \end{aligned}$$

(35)

Then $u \in BV({\varOmega },\nu )$ and $|D_\nu u|({\varOmega })\le C\Vert Q_\infty ^{1/2}\Vert _{{\mathcal {L}}(X)}\sqrt{\pi }/2$.

Proof

We divide the proof into two steps.

Step 1. Here we prove that for any $v \in C^1_b(X)$, it holds that

$$\begin{aligned}&\lim _{t \rightarrow 0^+}\frac{1}{\sqrt{t}}\int _{\varOmega }\int _X|v(e^{-t}x+\sqrt{1-e^{-2t}y})-v(x)|\mathrm{d}\gamma (y)\mathrm{d}\nu (x) \nonumber \\&\quad =\frac{2}{\sqrt{\pi }}\int _{{\varOmega }}|D v(x)|\mathrm{d}\nu (x). \end{aligned}$$

(36)

To this aim, we observe that

$$\begin{aligned}&K_v(t):=\int _{\varOmega }\int _X|v(e^{-t}x+\sqrt{1-e^{-2t}}y)-v(x)|\mathrm {d}\gamma (y)\mathrm {d}\nu (x)\\=&\int _{\varOmega }\int _X\left| \! \int _0^t\frac{\mathrm {d}}{\mathrm {d}r}v(e^{-r}x+\sqrt{1-e^{-2r}}y)\mathrm {d}r\right| \mathrm {d}\gamma (y)\mathrm {d}\nu (x)\\=&\!\!\!\int _{\varOmega }\!\!\int _X\!\Big |\! \int _0^t\!\!\!\langle D v(e^{-r}\!x\!+\!\sqrt{1\!-\!e^{-2r}}y),-e^{-r}\!x\!+\!\frac{e^{-2r}}{\sqrt{1\!-\!e^{-2r}}}y\rangle \mathrm {d}r\Bigr |\mathrm {d}\gamma (y)\mathrm {d}\nu (x)\\\le&\int _0^t \!\!\frac{e^{-r}}{\sqrt{1-e^{-2r}}} \cdot \\\cdot&\int _{\varOmega }\int _X {\left| {\left\langle D v(e^{-r}\!x\!+\!\sqrt{1\!-\!e^{-2r}}y),-\sqrt{1\!-\!e^{-2r}}x\!+\!e^{-r}\!y\right\rangle }\right| } \mathrm {d}\gamma (y)\mathrm {d}\nu (x). \end{aligned}$$

Now, for r fixed we perform the “Gaussian rotation”

$$\begin{aligned} (x,y)\mapsto R_r(x,y):=(e^{-r}x+\sqrt{1-e^{-2r}y},-\sqrt{1-e^{-2r}}x+e^{-r}y)=:(u,w) \end{aligned}$$

to get, thanks to the invariance of $\gamma$ under $R_r$,

$$\begin{aligned} K_v(t)&\le \int _0^t \frac{e^{-r}}{\sqrt{1-e^{-2r}}}\int _X\int _X |{\left\langle D v(u),w\right\rangle }|\cdot \\&\quad \cdot \chi _{{\varOmega }}(e^{-r}u- \sqrt{1- e^{-2r}}w)e^{-U(e^{-r}u-\sqrt{1- e^{-2r}}w)}\mathrm{d}\gamma (u)d\gamma (w)\\&=:\int _X\int _X f_v(t,u,w)\mathrm{d}\gamma (w)d\gamma (u). \end{aligned}$$

We claim that

$$\begin{aligned} \lim _{t \rightarrow 0^+}\frac{1}{\sqrt{t}}\int _X\int _X f_v(t,u,w)\mathrm{d}\gamma (w)d\gamma (u)=C\int _{\varOmega }|D v|\mathrm{d}\nu . \end{aligned}$$

Indeed, by the convexity of U there exist $z\in X$ and $a\in {{\mathbb {R}}}$ such that $U(x)\ge {\left\langle x,z\right\rangle }+a$, hence

$$\begin{aligned}&\frac{1}{\sqrt{t}}f_v(t,u,w)\le \frac{1}{\sqrt{t}}|D v(u)||w|e^{|{\left\langle z,u\right\rangle }|+|{\left\langle z,w\right\rangle }|+|a|}\int _{0}^t\frac{1}{\sqrt{2r}}\mathrm{d}r\\&\quad =\sqrt{2}|D v(u)||w|e^{|{\left\langle z,u\right\rangle }|+|{\left\langle z,w\right\rangle }|+|a|}\in L^1(X\times X, \gamma \otimes \gamma ), \qquad \;\, t \in (0,1) \end{aligned}$$

and, using De L’Hôpital’s rule, for almost every $(u,w)\in {\varOmega }\times X$

$$\begin{aligned} \lim _{t \rightarrow 0^+}\frac{f_v(t,u,w)}{\sqrt{t}}=\sqrt{2}\chi _{{\varOmega }}(u){\left\langle D v(u),w\right\rangle }e^{-U(u)}. \end{aligned}$$

So by the dominated convergence theorem we obtain

$$\begin{aligned}&\limsup _{t\rightarrow 0^+}\frac{1}{\sqrt{t}}\int _{\varOmega }\int _X|v(e^{-t}x+\sqrt{1-e^{-2t}y})-v(x)|\mathrm{d}\gamma (y)\mathrm{d}\nu (x) \nonumber \\&\quad \le \frac{2}{\sqrt{\pi }}\int _{\varOmega }|D v(u)|\mathrm{d}\nu (u) \end{aligned}$$

(37)

where we used that $\int _X|{\left\langle D v(u),w\right\rangle }|\mathrm{d}\gamma (w) = \sqrt{2/\pi }|D v(u)|$. Indeed, using the factorisation $\gamma = \gamma _1\otimes \gamma ^\perp$, where $\gamma _1$ is the 1-dimensional standard Gaussian measure on $E=\mathrm{span}Dv(u)$, we get

$$\begin{aligned}&\int _X|{\left\langle D v(u),w\right\rangle }|\mathrm{d}\gamma (w) = 2 \int _{\{w:{\left\langle D v(u),w\right\rangle }>0\}} {\left\langle D v(u),w\right\rangle } \mathrm{d}\gamma (w)\\&\quad =2 |D v(u)| \int _{E^\perp }\int _0^\infty t \mathrm{d}\gamma _1(t)d\gamma (w') = \sqrt{2/\pi }|D v(u)|. \end{aligned}$$

To conclude, consider the family of linear functionals $L_t: C_b(X\times X)\rightarrow {{\mathbb {R}}}$, $t \in (0,1)$

$$\begin{aligned} L_t\varphi =\frac{1}{\sqrt{t}}\int _{\varOmega }\int _X\varphi (x,y)(v(e^{-t}x+\sqrt{1-e^{-2t}}y)-v(x))\mathrm{d}\gamma (y)\mathrm{d}\nu (x). \end{aligned}$$

By (37) we get $\limsup _{t\rightarrow 0^+}{\left\| L_t\right\| }\le 2(\sqrt{\pi })^{-1}\Vert D v\Vert _{L^1({\varOmega },\nu )}$ and arguing as above

$$\begin{aligned} \lim _{t\rightarrow 0^+}L_t\varphi =\sqrt{2}\int _{\varOmega }\int _X\varphi (x,y){\left\langle D u(x),y\right\rangle }\mathrm{d}\gamma (y)\mathrm{d}\nu (x)=:L_0\varphi . \end{aligned}$$

So $L_t$ weakly$^*$ converges to $L_0$ as $t \rightarrow 0^+$ and, by lower semicontinuity of the norm we get (36):

$$\begin{aligned} {\left\| L_0\right\| }&=\frac{2}{\sqrt{\pi }}\int _{\varOmega }|D v(x)|\mathrm {d}\nu (x)\le \liminf _{t\rightarrow 0^+}{\left\| L_t\right\| }\\ {}&\le \limsup _{t\rightarrow 0^+} {\left\| L_t\right\| }\le \frac{2}{\sqrt{\pi }}\int _{\varOmega }|D v(x)|\mathrm {d}\nu (x). \end{aligned}$$

Step 2. For $u\in L^\infty (X, \nu )$, let $(u_j)_{j\in {{\mathbb {N}}}}\subseteq C^1_b(X)$ be such that $u_j\rightarrow u$ in $L^2(X, \nu )$, almost everywhere in X and satisfying (35) (thanks to the dominated convergence theorem). Using (36), (6) and (8) we have

$$\begin{aligned}&\lim _{t\rightarrow 0^+}\frac{K_{u_j}(t)}{\sqrt{t}} =\frac{2}{\sqrt{\pi }}\int _{{\varOmega }}|D u_j|\mathrm{d}\nu =\frac{2}{\sqrt{\pi }\Vert Q_\infty ^{1/2}\Vert _{{\mathcal {L}}(X)}}\int _{{\varOmega }}\Vert Q_\infty ^{1/2}\Vert _{{\mathcal {L}}(X)}|D u_j|\mathrm{d}\nu \nonumber \\&\quad \ge \frac{2}{\sqrt{\pi }\Vert Q_\infty ^{1/2}\Vert _{{\mathcal {L}}(X)}}\int _{{\varOmega }}|Q_\infty ^{1/2} D u_j|\mathrm{d}\nu =\frac{2}{\sqrt{\pi }\Vert Q_\infty ^{1/2}\Vert _{{\mathcal {L}}(X)}}\int _{{\varOmega }}|Q_\infty D u_j|_H\mathrm{d}\nu \nonumber \\&\quad =\frac{2}{\sqrt{\pi }\Vert Q_\infty ^{1/2}\Vert _{{\mathcal {L}}(X)}}\int _{{\varOmega }}|D_H u_j|_H\mathrm{d}\nu =\frac{2}{\sqrt{\pi }\Vert Q_\infty ^{1/2}\Vert _{{\mathcal {L}}(X)}}\int _{{\varOmega }}(T_{\varOmega }(\sigma )|D_H u_j|_H)\mathrm{d}\nu \nonumber \\&\quad \ge \frac{2}{\sqrt{\pi }\Vert Q_\infty ^{1/2}\Vert _{{\mathcal {L}}(X)}}e^{\sigma \lambda _1^{-1}} \int _{{\varOmega }}|D_H T_{\varOmega }(\sigma ) u_j|_H\mathrm{d}\nu \end{aligned}$$

(38)

for any $\sigma \in (0,1)$. Now, since the left-hand side of (38) is uniformly bounded from above by the constant C, the $L^1$-norm of $D_HT_{\varOmega }(\sigma )u_j$ is bounded as well by the same constant for every $j \in {{\mathbb {N}}}$ and $\sigma \in (0,1)$, i.e.

$$\begin{aligned} e^{\frac{\sigma }{\lambda _1}}\int _{{\varOmega }}|D T_{\varOmega }(\sigma ) u_j|\mathrm{d}\nu \le C\frac{\sqrt{\pi }}{2}\Vert Q_\infty ^{1/2}\Vert _{{\mathcal {L}}(X)},\qquad j\in {{\mathbb {N}}},\sigma >0. \end{aligned}$$

Thus, recalling that $D_H T_{\varOmega }(\sigma ) u_j$ converges to $D_H T_{\varOmega }(\sigma ) u$ in $L^1({\varOmega }, \nu )$ as $j\rightarrow \infty$ (see (7)), letting first $j \rightarrow \infty$ and then $\sigma \rightarrow 0^+$ and using formula (24) we get that $|D_\nu u|({\varOmega })\le C\Vert Q_\infty ^{1/2}\Vert _{{\mathcal {L}}(X)}\sqrt{\pi }/2$. $\square$

The following result is a quasi-converse of Theorem 5. In fact, we give a sufficient condition to have $P_\nu (E,\varOmega )<\infty$ in terms of the short-time behaviour of T(t) and not of $T_{\varOmega }(t)$, where T(t) is the semigroup generated by the operator L defined in (31) in $L^2(X,\nu )$.

Theorem 6

Under Hypotheses 1, 2, 3 and 4, if $E\in {{\mathcal {B}}}(X)$ and

$$\begin{aligned} C:=\liminf _{t \rightarrow 0^+}\frac{1}{\sqrt{t}}\Vert T(t)\chi _E- \chi _E\Vert _{L^1({\varOmega },\nu )}<\infty , \end{aligned}$$

(39)

then $P_\nu (E, {\varOmega })\le C\Vert Q_\infty ^{1/2}\Vert _{{\mathcal {L}}(X)}\sqrt{\pi }/2$.

Proof

Choosing $u=\chi _E$ in (35) and observing that

$$\begin{aligned} \int _{\varOmega }\left| \int _X f(x,y)\mathrm{d}\gamma (y)\right| \mathrm{d}\nu (x)=\int _{\varOmega }\int _X |f(x,y)|\mathrm{d}\gamma (y)\mathrm{d}\nu (x) \end{aligned}$$

for any f with constant sign, from Proposition 7 we deduce that if

$$\begin{aligned} L:=\liminf _{t \rightarrow 0^+}\frac{1}{\sqrt{t}}\Vert S(t)\chi _E- \chi _E\Vert _{L^1({\varOmega },\nu )}<\infty \end{aligned}$$

(40)

then $P_\nu (E, {\varOmega })\le L\Vert Q_\infty ^{1/2}\Vert _{{\mathcal {L}}(X)}\sqrt{\pi }/2$. Here S(t) is the Ornstein–Uhlenbeck semigroup in (1). To conclude we prove that condition (40) is equivalent to (39). From the variation-of-constants formula we deduce

$$\begin{aligned} (T(t)g)(x) = (S(t)g)(x) -\int _0^t (S(t-\sigma )\langle D_H U, D_H T(\sigma )g\rangle _H)(x) d\sigma , \end{aligned}$$

(41)

for every $g\in {\mathcal {F}}C_b(X)$, $\nu$-a.e. $x\in X$ and any $t\ge 0$. To prove (41) it suffices that the map $\sigma \mapsto S(t-\sigma )\langle D_H U, D_H T(\sigma )g\rangle _H$ belongs to $L^1((0,t))$ for any $t>0$. To this aim, let us observe that

$$\begin{aligned} \int _X\int _0^t S(t-\sigma )\langle D_H U, D_H T(\sigma )g\rangle _H d\sigma \mathrm{d}\nu <\infty \end{aligned}$$

for any $g\in {\mathcal {F}}C_b(X)$. Indeed, the Hölder inequality and the contractivity of S(t) in $L^2(X,\gamma )$ allow us to write

$$\begin{aligned} \int _X&\int _0^t S(t-\sigma )\langle D_H U, D_H T(\sigma )g\rangle _H d\sigma \mathrm{d}\nu \nonumber \\&= \int _0^t\int _X S(t-\sigma )\langle D_H U, D_H T(\sigma )g\rangle _H \mathrm{d}\nu d\sigma \nonumber \\&\le \int _0^t\Vert S(t-\sigma )\langle D_H U, D_H T(\sigma )g\rangle _H\Vert _{L^1(X, \nu )} d\sigma \nonumber \\&\le \Vert e^{-U}\Vert _{L^2(X,\gamma )} \int _0^t\Vert S(t-\sigma )\langle D_H U, D_H T(\sigma )g\rangle _H\Vert _{L^2(X, \gamma )} d\sigma \nonumber \\&\le \Vert e^{-U}\Vert _{L^2(X,\gamma )}\int _0^t\Vert \langle D_H U, D_H T(\sigma )g\rangle _H\Vert _{L^2(X, \gamma )} d\sigma \nonumber \\&\le \sqrt{K_2}\Vert e^{-U}\Vert _{L^2(X,\gamma )}\Vert g\Vert _\infty \Vert D_H U\Vert _{L^2(X,\gamma ;H)}\int _0^t \sigma ^{-1/2} \mathrm{d}s \nonumber \\&= 2\sqrt{K_2 t}\Vert e^{-U}\Vert _{L^2(X,\gamma )}\Vert g\Vert _\infty \Vert D_H U\Vert _{L^2(X,\gamma ;H)} , \end{aligned}$$

(42)

where in the last line we used estimate (7) which holds true even in the case ${\varOmega }=X$ and $T_{\varOmega }(t)$ replaced by T(t). Hence, formula (41) follows.

Now, integrating (41) in ${\varOmega }$ with respect to $\nu$ yields

$$\begin{aligned} \Vert S(t)\chi _E-\chi _E\Vert _{L^1({\varOmega }, \nu )}- H(t)&\le \Vert T(t)\chi _E-\chi _E\Vert _{L^1({\varOmega }, \nu )} \nonumber \\&\le \Vert S(t)\chi _E-\chi _E\Vert _{L^1({\varOmega }, \nu )}+ H(t) \end{aligned}$$

(43)

for any $t>0$ with

$$\begin{aligned} H(t):= \left| \int _{X}\int _0^t S(t-s)\langle D_H U,D_H T(s)\chi _E\rangle _H \mathrm{d}s \mathrm{d}\nu \right| , \qquad \;\, t>0. \end{aligned}$$

Using estimate (42) with $g=\chi _E$ we infer that $\limsup _{t\rightarrow 0^+}\frac{H(t)}{\sqrt{t}}<\infty$. This last estimate, together with (43), prove that (40) is equivalent to (39) and the proof is complete. $\square$