On functions of bounded variation on convex domains in Hilbert spaces

We study functions of bounded variation (and sets of finite perimeter) on a convex open set Ω⊆X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varOmega }\subseteq X$$\end{document}, 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.


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 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 γ is a centered and nondegenerate Gaussian measure on X and u belongs to the Orlicz space L(log L) 1/2 (X, γ), then u ∈ BV (X, γ) if, and only if, lim inf where D H is the gradient operator along the Cameron-Martin space H (see Section 1) and S(t) is the classical Ornstein-Uhlenbeck semigroup defined via the Mehler formula (see (1.1)).This 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, γ) 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 ν := e −U γ in a Hilbert space X, where U : X → R is convex and sufficiently regular, and consider an open convex domain Ω ⊆ X.After introducing the Cameron-Martin space H and the Malliavin gradient D H along it, we define the form (u, v) → Ω D H u, D H v H dν on the appropriate Sobolev spaces.The perturbed Ornstein-Uhlenbeck operator L Ω is then defined in the usual variational way, and it is the generator of an analytic, strongly continuous and contraction semigroup T Ω (t) in L p (Ω, ν), for 1 < p < ∞.
For this 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 Ω ⊆ X of BV (X, γ) functions and they characterise the finiteness of their total variation in Ω in terms of the Neumann Ornstein-Uhlenbeck semigroup defined in Ω.
Following the ideas in [2], we define the BV (Ω, ν) space through an integration by parts formula against suitable Lipschitz functions.Then we show that the functions u of bounded variation in Ω with respect to ν can be characterised by the finiteness of the limit of D H T Ω (t)u L 1 (Ω,ν;H) as t → 0 + .The proof of this result relies on a commutation formula between the semigroup T Ω (t) and the gradient operator along H (see Proposition 2.6).This result was already known in the case of the whole space (see [12]).Here, by means of the crucial pointwise gradient estimate (1.4) and suitable penalisations Φ ε of U outside Ω based on the distance function from Ω along H (here is a first point where the convexity of Ω comes into the play) and the penalisation ν ε := e −Φε γ of the measure ν, see Subsection 1.1, we are able to let ε to 0 + and to to come back to Ω.
Finally we provide a necessary condition in order that a set E is of finite perimeter in Ω with respect to ν (i.e., χ E ∈ BV (Ω, ν)).This condition is given in terms of the short-time behaviour of the Ornstein-Uhlenbeck content T Ω (t)χ E − χ E L 1 (Ω,ν) as t → 0 + .Further, a sufficient condition in terms of a related quantity is also shown.and a sufficient condition in terms of a related quantity.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].

Hypotheses and preliminaries
Let H 1 and H 2 be two real Hilbert spaces with inner products •, • H1 and •, • H2 respectively.We denote by B(H 1 ) the σ-algebra of Borel subsets of H 1 and by , for every x ∈ H 1 there exists a unique k ∈ H 1 such that Df (x)(h) = h, k H1 , h ∈ H 1 and we set Df (x) := k.Let X be a separable Hilbert space, with inner product •, • and norm | • |.Let B ∈ L(X) (the set of bounded linear operators from X to itself).We say that B is non-negative if Bx, x ≥ 0 for every x ∈ X and positive if Bx, x > 0 for every x ∈ X \ {0}.We recall that a non-negative and self-adjoint operator B ∈ L(X) is a trace class operator whenever Tr(B) := ∞ n=1 Be n , e n < ∞ for some (and hence, every) orthonormal basis (e n ) n∈N of X.
Let γ be a nondegenerate Gaussian measure on X with mean zero and covariance operator Q ∞ := −QA −1 , where the operators Q and A satisfy the following assumptions.
Hypotheses 1. (i) Q ∈ L(X) is a self-adjoint and non-negative operator with Ker Q = {0}; (ii) A : D(A) ⊆ X → X is a self-adjoint operator satisfying Ax, x ≤ −ω|x| 2 for every x ∈ D(A) and some positive ω; Under Hypotheses 1(i)-(iii), the measure γ is well defined and the Ornstein-Uhlenbeck semigroup defined via the Mehler formula is symmetric in L 2 (X, γ).We fix an orthonormal basis (v k ) k∈N of X such that where (λ k ) k∈N is the decreasing sequence of eigenvalues of Q ∞ .Under Hypothesis 1(iv), the Cameron-Martin (H, |•| H ) k , is a Hilbert space compactly and densely embedded in X (see [8] and [13] for further details).The sequence (e k ) k∈N , where is the generator of a contractive and strongly continuous semigroup e −tQ −1 ∞ on X (on H, respectively), see [14,Proposition p. 84]).If Y is a Banach space with norm • Y , a function F : X → Y is said to be H-Lipschitz continuous if there exists a positive constant C such that as |h| H → 0.
In such a case we set D H f (x 0 ) := ℓ and D i f (x 0 ) := D H f (x 0 ), e i H for any i ∈ 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 B ∈ H 2 such that In such a case we set where the equality must be understood as holding in H.For any k ∈ N ∪ {∞}, we denote by FC k b (X), the space of cylindrical C k b functions, i.e., the set of functions f : functions with finite rank.The Sobolev spaces in the sense of Malliavin D 1,p (X, γ) and D 2,p (X, γ) with p ∈ [1, ∞), are defined as the completions of the smooth cylindrical functions FC ∞ b (X) in the norms This is equivalent to considering the domain of the closure of the gradient operator, defined on smooth cylindrical functions, in L p (X, γ) (see [8,Section 5.2]).Let U : X → R satisfy the following assumptions.
The convexity of the function U guarantees that U is bounded from below by a linear function, therefore it decreases at most linearly and by Fernique theorem (see [8, Theorem 2.8.5]) e −U belongs to L 1 (X, γ).Then we can consider the finite log-concave measure It is obvious that γ and ν are equivalent measures, hence saying that a statement holds γa.e. is the same as saying that it holds ν-a.e.Moreover as U ∈ ∩ q≥1 D 1,q (X, γ), the operator and the space D 1,p (X, ν), 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, ν), p ∈ (1, ∞) (for more details see [1,9,16]).The Gaussian integration by parts formula In what follows Ω denotes an open subset of X.In this case, the spaces D 1,p (Ω, ν) and D 2,p (Ω, ν), p ∈ (1, ∞), can be defined in a similar way as in the whole space, thanks to the following result (see [6,Proposition 1.4]).
are defined in a similar way, replacing smooth cylindrical functions with H-valued smooth cylindrical functions with finite rank.We recall that if F ∈ D 1,p (Ω, ν; H), then D H F (x) belongs to H 2 for a.e.x ∈ Ω.We denote by p ′ the conjugate exponent to p ∈ (1, ∞).

Perturbed Ornstein-Uhlenbeck semigroup on convex domains
In order to consider the initial boundary value problems defined in Ω we define the distance function along x ∈ 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 1.2.If Ω ⊆ X be an open convex set, then d 2 Ω is H-differentiable and its Malliavin derivative is H-Lipschitz with H-Lipschitz constant less than or equal to 2, i.e., We require some further regularity on d 2 Ω .
Hypotheses 3. Let Ω be an open convex subset of X such that ν(∂Ω) = 0 and D 2 H d 2 Ω is H-continuous γ-a.e. in X, i.e., for γ-a.e.x ∈ X we have We consider the semigroup T Ω (t) on L 2 (Ω, ν) and its generator L Ω : functions that relies on reduction to a finite (say n-) dimensional space and on a ε-penalisation argument.Accordingly, the approximation depends on two parameters n and ε.More precisely, we consider the function and the measure ν ε given by e −Φε γ.Next, we consider the operator L ε on the whole X defined as , and the semigroup T ε (t) generated by L ε in L 2 (X, ν ε ).We point out that L ε acts on smooth cylindrical functions ϕ as follows Now we recall a useful approximation result whose proof can be found in [6, Theorem 2.8].
In addition, if f ∈ D 1,2 (X, ν ε ) then the sequence (f n ) can be chosen in a way that (ii) For any f ∈ L 2 (Ω, ν) there exists an infinitesimal sequence (ε n ) n∈N such that T εn (t) f weakly converges to T Ω (t)f in D 1,2 (Ω, ν), where f is any L 2 -extension of f to X.
We collect some properties of T Ω (t), see [ We point out that the results in Proposition 1.5 continue to hold if we replace Ω, ν and T Ω (t) by X, ν ε and T ε (t), respectively.

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 |•| Y .We recall that in separable spaces the σ-algebra B(X) is generated by the family of the cylindrical sets (see e.g.[25]).Denote by M(Ω; Y ) the set of Borel Y -valued measures on Ω.If Y = R then we write M(Ω).The total variation of µ ∈ M(Ω; Y ) is the positive Borel measure Let Lip c (Ω; Y ) be the set of bounded Lipschitz continuous Y -valued functions g : Ω → Y such that dist(supp g, X Ω) > 0 and define the space BV (Ω, ν) as follows.
Definition 1.6.Let Ω be an open subset of X.We say that a function f ∈ L 2 (Ω, ν) is of bounded variation in Ω, and we write f ∈ BV (Ω, ν), if there exists a measure µ ∈ M(Ω; H) such that for every g ∈ Lip c (Ω) and h ∈ H, where ∂ * h denotes, up to the sign, the adjoint in L 2 (Ω, ν) of the partial derivative along h ∈ H.In this case we set D ν f := µ.
As in the finite dimensional case, one can characterise functions of bounded variation by their total variation.Definition 1.7.Let Ω be an open subset of X and u ∈ L 2 (Ω, ν).We define the variation of u in Ω by .
When Ω = X, in the two definitions above we can consider Lip b (X) and Lip b (X; F ) respectively, as test functions spaces.
As announced, in [2, Theorem 5.7] it has been proved that u ∈ BV (Ω, ν) if and only if V ν (u, Ω) is finite.Moreover, in this case (1.1) Finally we say that a Borel subset E of X is of finite perimeter in Ω with respect to ν, whenever the function χ E belongs to BV (Ω, ν).In this case we denote by P ν (E, Ω) the total variation of χ E in Ω.

A De Giorgi type characterisation
The main result of this section is the De Giorgi type characterisation of BV (Ω, ν) functions in Theorem 2.8, which relies on a "quasi-commutative" formula between the semigroup T Ω (t) and the H-gradient operator D H ; here estimate (1.5) 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, ν ε ; H).
Definition 2.1.We denote by D 1,2 (X, ν ε ; H) the domain of the closure of the operator where {e i | i ∈ N} is an orthonormal basis of H and ) for every i = 1, . . ., n.In an analogous way we define the space D 1,2 (Ω, ν; H).
We first show a vector-valued version of Theorem 1.4.Let L ε in L 2 (X, ν ε ; H) be the operator defined via the quadratic form by In the same way we define the operator L Ω in in L 2 (Ω, ν; H).We recall that by [14, Moreover for every t > 0, if F ∈ L 2 (X, ν ε ; H) (L 2 (Ω, ν; H), respectively), and it is such that The above identities hold ν ε -a.e. in X (ν Ω -a.e. in Ω, respectively).
Proof.We only show the results for L ε and Thus, by the uniqueness of the solution of the Cauchy problem associated with D t − L ε in L 2 (X, ν ε ), it follows that (T ε (t)F ) j = T ε (t)f j for any t > 0. The arbitrariness of j ∈ N concludes the proof.
Remark 2.3.According to the definition of T ε (t) and T Ω (t) it is immediately seen that for every F ∈ L 2 (X, and Moreover, taking into account that the semigroups T Ω (t) and T ε (t) act component by component, we can obtain a vector-valued version of Theorem 1.4.
(ii) For any F ∈ L 2 (Ω, ν; H) there exists an infinitesimal sequence For every i ∈ N, by Theorem 1.4(i), there exists (f Observe that (2.3) follows immediately from (2.4).Now fix i, n ∈ N and consider Consider the vector field ki e i .We claim that (F n ) is the sequence we are looking for.Indeed F n belongs to L 2 (X, ν ε ; H) for any n ∈ N. Let n 0 ∈ N be such that L 2 (X,νε) ≤ η/2 and let n ≥ n 0 such that 1/n < η/2.We have In a similar way we can prove the other statements.
(ii) is an immediate consequence of Proposition 2.2 and Theorem 1.4(ii).
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 S(t)D H f holds true for any t ≥ 0. Let us prove a (quasi) commutation formula between T Ω (t) and D H , under the following additional assumption.
where K is a positive constant and γ n denotes the n-dimensional Gaussian measure, image of γ under the projection on span {v 1 , . . ., v n }.To conclude, observe that there exists n ∈ N such that the right-hand side of (2.5) is finite.
Proof.In order to prove (2.6) we show that for any f ∈ Lip c (Ω), G ∈ C b (Ω; H) and t > 0. By performing slight changes in [12, Appendix A] we get ν ε -a.e. in X for any g ∈ Lip b (X) and ε > 0, where T ε (t) is the semigroup introduced in Subsection 1.1.Now, let f ∈ Lip c (Ω) and f be the trivial extension to zero of f in the whole space X.Clearly, f belongs to Lip b (X) and (2.8) holds true with g replaced by f .Consequently, multiplying (2.8) by the function G and integrating on Ω with respect to ν yield where in the last line we used the Fubini-Tonelli theorem.
We split the proof of (2.7) in two steps.
Step 1.We argue by approximation on the last terms in (2.9) and (2.7).For every ε, s > 0 we fix a Borel measurable version of D H T Ω (s)f and D H T ε (s) f in L 2 (Ω, ν; H) and L 2 (X, ν ε ; H), respectively.Consider the function Observe that the map x → Γ ε (s, x) is an extension of D H T Ω (s)f to the whole X.Thus, by Theorem 2.4 there is a sequence ε n ↓ 0 such that for every η > 0 the function is bounded in L 2 (Ω, ν; H).Indeed by the contractivity of T εn (t) in the space L 2 (X, ν εn ; H), the fact that U ≡ Φ ε on Ω and estimate (1.5) we have where in the last line we used the contractivity of T η (t) and T Ω (t) in L ∞ and the fact that ν εn (X) ≤ ν(X) for any n ∈ N.So there exists M > 0 large enough so that the family in (2.10) is contained in B(0, M ), the ball of L 2 (Ω, ν; H) with center 0 and radius M .Recall that every bounded subset of L 2 (Ω, ν; H) is weakly metrisable (see [15, Proposition 3.106]) and let ρ : B(0, M ) × B(0, M ) → R be a metric such that the topology generated by ρ and the weak topology in B(0, M ) coincide.Now we use a diagonal argument to pass to the limit in (2.9).Let n 1 ∈ N such that for every n ≥ n 1 it holds where Γ j (s, x) = Γ j −1 (s, x) for any s > 0 and x ∈ X.Now assume that n 1 , . . ., n k are already constructed and consider n k+1 > n k be such that for every Step 2. To complete the proof, we replace ε in (2.9) by a sequence ε m ↓ 0 such that step 1 and Theorems 1.4, 2.4 apply.Let us show that we can take the limit as m → ∞.Indeed, from Theorem 1.4 it follows that for any f ∈ L 2 (Ω, ν), T εm (t) f weakly converges (up to a subsequence) to and by the analogous vector-valued result (see Theorem 2.4, (2.1) and again (1.4)) To conclude we have to prove that the last term in the right hand side of (2.9) converges to the last term in the right hand side of (2.7).
Let us estimate I 1 .Using that Φ m ≡ U on Ω for every m ∈ N, formula (2.1) and the invariance property of T m with respect to ν m := ν εm we have . (2.11) The right hand side of (2.11) converges to zero as m → ∞: indeed, D H T m (s) f converges pointwise ν-almost everywhere in Ω to D H T Ω (s)f .Furthermore, by Proposition 1.5 we have that ν-a.e. in Ω So by the dominated convergence theorem we get that I 1 (m) vanishes as m → ∞.Now, using similar arguments we can estimate I 2 (m) as follows . (2.12) Now observe that the right hand side of (2.12) vanishes as m → ∞.Indeed the function Ω identically vanishes in Ω and converges pointwise to 0 ν-almost everywhere in X \ Ω as m → ∞.Furthermore let observe that the function (0, H2 .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.
Theorem 2.8.Assume Hypotheses 1, 2, 3 and 4 hold true and let u ∈ L 2 (Ω, ν).The following statement are true: Hence, u ∈ BV (Ω, ν) iff lim Proof.(i) follows from the strong continuity of T Ω (t) in L 1 (Ω, ν), see Proposition 1.5(i), and the lower semicontinuity of the norm (1.1), which imply To prove (ii) we write the L 1 -norm of the gradient of T Ω (t)u by duality, as Taking into account that, for any F ∈ Lip b (Ω; H) we get for any t > 0 where C i (i = 1, 2) are the positive functions in Corollary 2.7.Thus, taking the limsup as t → 0 + in (2.16) we get lim sup and the proof is complete.
It follows from Theorem 2.8 that functions in BV (Ω, ν) may be approximated in variation by smooth functions.This result was already known in infinite dimension when Ω = X and T Ω (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 Ω. Proposition 2.9.Under Hypotheses 1, 2, 3 and 4, for any f ∈ BV (Ω, ν) there exists a sequence (2.17) Proof.Consider the semigroup T Ω (t) generated in L 2 (Ω, ν) by the operator L Ω defined in (1.1).It is known that for any f ∈ L 2 (Ω, ν) the function T Ω (t)f belongs to D where C denotes the interior of C. Estimate (2.20) yields the claim.
We conclude this section showing that estimate (1.5) and the previous approximation result allow to improve estimate (2.16) obtaining (2.21).

Sets of finite perimeter in Ω
This section is devoted to provide some sufficient and necessary conditions in order that a Borel set E ⊆ X have finite perimeter in Ω.We consider also the case of BV (Ω, ν) functions and Ω = X.There are three semigroups involved: beside T Ω (t), we consider the Ornstein-Uhlenbeck semigroup S(t) generated in L 2 (X, γ) by the realisation of the operator and the semigroup T (t) generated in L 2 (X, ν) by the realisation of the operator Recall that S(t) admits a pointwise representation by means of the Mehler formula (1.1).
Step 1.Here we prove that for any v ∈ C 1 b (X), it holds that for any f with constant sign, from Proposition 3.3 we deduce that if √ π/2.Here S(t) is the Ornstein-Uhlenbeck semigroup in (1.1).To conclude we prove that condition (3.10) is equivalent to (3.9).From the variationof-constants formula we deduce (T (t)g)(x) = (S(t)g)(x) − for every g ∈ FC b (X), ν-a.e.x ∈ X and any t ≥ 0. To prove (3.11) it suffices that the map σ → S(t − σ) D H U, D H T (σ)g H belongs to L 1 ((0, t)) for any t > 0. To this aim, let us observe that where in the last line we used estimate (1.4) which holds true even in the case Ω = X and T Ω (t) replaced by T (t).Hence, formula (3.11)Using estimate (3.12) with g = χ E we infer that lim sup t→0 + H(t) √ t < ∞.This last estimate, together with (3.13), prove that (3.10) is equivalent to (3.9) and the proof is complete.

Remark 1 . 3 .
As stated in [6, Remark 1.7] there is a rather large class of subsets of X satisfying Hypothesis 3.For instance if ∂Ω is (locally) a C 2 -embedding in X of an open subset of a hyperplane in X and ν(∂Ω) = 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

Hypotheses 4 . 2 Ω
The map (d Ω ) −2 D 2 H d H2 belongs to L 2 (X, ν).Remark 2.5.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 Ω is the unit ball B X centered at zero.The other examples can be discussed in a similar fashion.Observe that, by Proposition 1.2,