BV functions in abstract Wiener spaces

Transcription

1 BV functions in abstract Wiener spaces Luigi Ambrosio, Michele Miranda Jr, Stefania Maniglia, Diego Pallara. June 12, 2009 Abstract Functions of bounded variation in an abstract Wiener space, i.e., an infinite dimensional Banach space endowed with a Gaussian measure and a related differential structure, have been introduced by M. Fukushima and M. Hino using Dirichlet forms, and their properties have been studied with tools from analysis and stochastics. In this paper we reformulate, in an integralgeometric vein and with purely analytical tools, the definition and the main properties of BV functions, and investigate further properties. 1 Introduction The spaces BV of functions of bounded variation in Euclidean spaces are by now a classical setting where several problems, mainly (but not exclusively) of variational nature, find their natural framework. Recently, generalizations in metric measure spaces have been studied, but mostly under the hypothesis that the measure is doubling, which is not the case when dealing with probability measures in vector spaces, not even locally in infinite dimensions. However, there are several motivations for studying BV functions in this context, i.e., Banach spaces, also of infinite dimensions, endowed with a probability measure. Among them, let us quote isoperimetric inequalities and mass concentration, see [20], [21], infinite dimensional analysis and semigroups (see e.g. [9], [11]). Moreover, the importance of generalizing the classical notion of perimeter and variation has been pointed out in several occasions by E. De Giorgi: we refer to [13], where the infinite dimensional context is explicitly mentioned. BV functions in an abstract Wiener space, i.e., a Banach space endowed with a Gaussian measure γ and a related differential structure, have been defined by M. Fukushima in [18], M. Fukushima and M. Hino in [19], relying upon Dirichlet forms theory. The starting point of these papers has been a characterization of sets with finite perimeter in finite dimensions in terms of the behaviour of suitable stochastic processes (see [17]), and in fact some arguments in [18], [19] come from stochastics. In this infinite-dimensional context, the main difficulty arises from the fact that the measures appearing in the integration by parts formula can t be built by a direct approximation by smooth functions, since Riesz theorem is not available and the dual of C b () strictly contains the class of signed measures; so, looking for instance at the approximation of u by Scuola Normale Superiore, Piazza dei Cavalieri,7, Pisa, Italy, e mail: l.ambrosio@sns.it Dipartimento di Matematica, Università di Ferrara, via Machiavelli, 35, 44100, Ferrara, Italy, e mail: michele.miranda@unife.it Dipartimento di Matematica Ennio De Giorgi, Università del Salento, C.P.193, 73100, Lecce, Italy, e mail: stefania.maniglia@unisalento.it, diego.pallara@unisalento.it 1

2 the Ornstein Uhlenbeck semigroup T t u as t 0, one has to prove tightness of the family of measures H T t uγ as t 0. In the above mentioned papers this difficulty is overcome (after a reduction to nonnegative functions u) by looking at the Dirichlet form E u (v, v) = u Hv 2 Hdγ and applying the capacity theory of Dirichlet forms. One of the aims of this paper is to study BV functions in abstract Wiener spaces with tools closer to those used in the Euclidean case. We analyse the connections between the distributional notion of (vector-valued) measure gradient and the approximation by smooth functions, as well as the relevant properties of the Ornstein-Uhlenbeck semigroup. Our methods rely basically on an integralgeometric approach, a viewpoint which has already proved to be very useful in the finite-dimensional theory (see also [16] for a recent contribution in the same direction in Wiener spaces). This approach allows to build directly the measure distributional derivative and to obtain the tightness property only as a consequence. Some of the main results of this paper have been presented, together with several open problems, in [2]. The paper is organized as follows: in Section 2 we describe the Wiener space setting and the tools useful to rephrase in this framework the characterizations of total variation that are known in the Euclidean case. A more detailed comparison with the Euclidean case is presented in [2]. In Section 3 we define BV functions in Wiener spaces and discuss their basic properties. In Section 4 we provide a complete characterization of BV functions in terms of integration by parts formulae and approximation by smooth functions. It is known that neither Sobolev nor BV spaces are compactly embedded in L p spaces, but our integral-geometric viewpoint provides new natural compactness criteria, both in the Sobolev and BV classes. In Section 5, in the case when the Wiener space is Hilbert, we compare Sobolev and BV classes with those developed by Da Prato and collaborators, proving compactness of these stronger Sobolev and BV classes. 2 Wiener space setting In this section we describe our setting: given an (infinite dimensional) separable Banach space, we denote by its norm and by B (x, r) = {y : y x < r the open ball centred at x and with radius r > 0. denotes the topological dual, with duality,. Given the elements x 1,..., x m in, we denote by Π x 1,...,x m : Rm the map (1) Π x 1,...,x m x = ( x, x 1,..., x, x m ), also denoted by Π m : R m if it is not necessary to specify the elements x 1,..., x m. The symbol FCb k () denotes the space of k times continuously differentiable cylindrical functions with bounded derivatives up to the order k, that is, u FCb k() if u(x) = v(π mx) for some v Cb k(rm ). We divide this section in some subsections; first of all we recall some notion of measure theory, with particular emphasis on the infinite dimensional (i.e., non locally compact) setting, then we pass to the definition and description of abstract Wiener spaces. In the third subsection we discuss the integration by parts formula and recall the definitions of gradient and divergence. Finally, we introduce Sobolev classes and the Ornstein-Uhlenbeck semigroup together with some of their basic properties. 2

3 2.1 Infinite dimensional measure theory We denote by B() the Borel σ-algebra; since is separable, B() is generated by the cylindrical sets, that is by the sets of the form E = Π 1 m B with B B(R m ), see [24, Theorem I.2.2]; this fact remains true even if we fix a sequence (x i ) which separates the points in and use only elements from that sequence to generate the maps Π m. We shall make later some special choices of (x i ), induced by a Gaussian probability measure γ in. We also denote by M (, Y ) the set of countably additive measures on with finite total variation with values in a Hilbert space Y, M () if Y = R. We denote by µ the total variation measure of µ, defined by { (2) µ (B) := sup µ(b h ) Y : B = B h, h=1 for every B B(), where the supremum runs along all the countable disjoint unions. Notice that, using the polar decomposition, there is a unit µ -measurable vector field σ : Y such that µ = σ µ, and then the equality { µ () = sup σ, φ d µ, φ C b (, Y ), φ(x) Y 1 x, holds, where, denotes the duality between Y and Y. Note that, by the Stone- Weierstrass theorem, the algebra FCb 1 () of C1 cylindrical functions is dense in C(K) in sup norm, since it separates points, for all compact sets K. Since µ is tight, it follows that FCb 1() is dense in L1 (, µ ). Arguing componentwise, it follows that also the space FCb 1(, Y ) of cylindrical functions with a finite-dimensional range is dense in L 1 (, µ, Y ). As a consequence σ can be approximated in L 1 (, µ, Y ) by a uniformly bounded sequence of functions in FCb 1(, Y ), and we may restrict the supremum above to these functions only to get { (3) µ () = sup σ, φ d µ, φ FCb 1 (, Y ), φ(x) Y 1 x. We now recall a tightness criterion and we include its proof for the reader s convenience. Lemma 2.1. Let (σ n ) M + () be a bounded sequence, σ M + () and assume that lim sup n σ n () σ(), while h=1 lim inf n σ n(a) σ(a) for all A open. Then (σ n ) is tight and σ n σ in the duality with C b (). Proof. Let (x i ) a dense sequence and ε > 0 be fixed. We claim that for all k 1 there exists N = N(k) such that ( N sup σ n \ B 1/k (x i ) ) < ε2 k. n i=1 It this property holds, the totally bounded and closed set K ε := k N(k) 1 B 1/k (x i ) satisfies sup n σ n ( \ K ε ) < ε, proving the tightness of (σ n ). 3

4 We prove the claimed property by contradiction, assuming that for some k there exist n(l) such that σ n(l) ( \ l i=1 B 1/k(x i )) > ε2 k for all l 1. Obviously n(l) as l and for any l 0 we have σ ( l0 i=1 B 1/k (x i ) ) lim inf σ ( l0 n(l) B 1/k (x i ) ) lim inf σ ( l n(l) B 1/k (x i ) ) l l i=1 i=1 lim sup σ n () ε2 k σ() ε2 k. n Letting l 0 gives a contradiction, since l0 1 B 1/k(x i ). The last statement is a simple consequence of the Cavalieri formula, taking into account that σ n (E) σ(e) for all Borel sets E with σ( E) = 0, and that σ({u = t) = 0 with at most countably many exceptions for all u C b (). Finally, let us define the sup of (the total variation of) an arbitrary family of measures {µ α : α I by setting { µ α (A) = sup µ αn (A n ), α I where the supremum runs along all the countable pairwise disjoint partitions A = n A n and all the choices of the sequence (α n ) I. 2.2 The abstract Wiener space Assume that a nondegenerate centred Gaussian measure γ is defined on. This means that γ is a probability measure and for all x the law x # γ is a centred Gaussian measure on R, that is, the Fourier transform of γ is given by { ˆγ(x ) = exp{ i x, x dγ(x) = exp 1 2 Qx, x, x, where Q L(, ) is the covariance operator. The nondegeneracy hypothesis means that γ is not concentrated on a proper subspace of, in terms of Q this means that Qx, x > 0 for x 0. The covariance operator is a symmetric and positive operator uniquely determined by the relation (4) Qx, y = x, x x, y dγ(x), x, y ; we also write N (0, Q) for γ. The fact that the operator Q defined by (4) is bounded is a consequence of Fernique s Theorem (see e.g. [7, Theorem 2.8.5]), asserting that (5) exp{α x 2 dγ(x) < if and only if n=1 { α 1 > σ := sup Qx, x 1/2 : x, x 1 ; as another consequence of this we also get that any x defines a function x x, x that belongs to L p (, γ) for all p 1. In particular, we can think of any x as a 4

5 continuous element of L 2 (, γ). Let us denote by R : L 2 (, γ) the embedding, R x (x) = x, x. The space H given by the closure of R in L 2 (, γ) is called the reproducing kernel of the Gaussian measure γ and obviously R turns out to be dense in it. The above definition is motivated by the fact that if we consider the operator R : H whose adjoint is R, then (6) Rĥ = ĥ(x)xdγ(x), where the integral is understood in Bochner s sense. As a consequence Q = RR : RR x, y = [R x, R y ] H = x, x x, y dγ(x) = Qx, y. It can be easily proved that R is injective. In addition, the operator R is compact and even more, i.e., it belongs to the ideal γ(h, ) of γ-radonifying, or Gaussian-Radonifying operators, see e.g. [22]. This remark shows that another presentation is possible: one can start with R γ(h, ) for some separable Hilbert space H and construct a Gaussian measure γ whose covariance operator is Q = RR. In any case, the measure γ built with this construction is concentrated on the separable subspace of defined as the closure of RH in. The space H = RH is called the Cameron-Martin space; it is a separable Hilbert space with inner product defined by [h 1, h 2 ] H = [ĥ1, ĥ2] H for all h 1, h 2 H, where h i = Rĥi, i = 1, 2. It is a dense subspace of and the embedding of (H, H ) in (, ) is compact since R is compact. In addition, γ(h) = 0 if is infinite-dimensional [7, Theorem 2.4.7], while H = if is finite-dimensional. With this notation, the Fourier transform of the Gaussian measure γ reads { ˆγ(x ) = exp 1 2 ˆx 2 H, x, where ˆx = R x. Using the embedding R H, we shall say that a family {x j of elements of is orthonormal if the corresponding family {R x j is orthonormal in H. Starting from a sequence (yj ) in whose image under R is dense in H, we may construct an orthonormal basis (R x j ) in H (by the Gram-Schmidt procedure), hence h j = Qx j = RR x j provide an orthonormal basis of H. Set also H m = span{h 1,..., h m, and define m = ker Π x 1,...,x m and m the (m-dimensional) complementary space. Since the variables R x j are Gaussian and uncorrelated, they are independent, hence the image γ m of γ under Π x 1,...,x is a m standard Gaussian in R m ; in addition it can be proved that we have a product decomposition γ = γ m γm of the measure γ, with γm Gaussian. Since R is dense in H the following proposition is easily estabilished by approximation: Proposition 2.2. Let ĥ1,..., ĥm be in H. Then the law of the variable x (ĥ1,..., ĥm) under γ is Gaussian. If ĥi are orthonormal, the law is the standard Gaussian γ m in R m. 5

6 One more property of Gaussian measures we shall use is rotation invariance, i.e., if ϱ : is given by ϱ(x, y) = (cos ϑx + sin ϑy, sin ϑx + cos ϑy) for some ϑ R, then ϱ # (γ γ) = γ γ. We shall use, in particular, the following equality: (7) u(cos ϑx + sin ϑy)dγ(x)dγ(y) = u(x)dγ(x), u L 1 (, γ), which is obtained by the above relation by integrating the function u 1 on. For every function u L 1 (, γ) we define its canonical cylindrical approximations E m u as the conditional expectations relative to the σ-algebras generated by { x, x 1,..., x, x m, denoted by B m (), i.e., (8) udγ = E m udγ, A B m (). A A Then, E m u u in L 1 (, γ) and γ-a.e. (see e.g. [7, Corollary 3.5.2]). More explicitly, we have E m u(x) = u(π m x + (I Π m )y)dγ(y) = u(π m x + y )dγm(y ), where Π m is the projection onto m. Notice that E m u is invariant under translations along all the vectors in m, hence we may write E m u(x) = v(π m x) for some function v, and, with an abuse of notation, we may write E m u(x m ) instead of E m u(x). Finally, let us recall the Cameron Martin Theorem: the shifted measure m γ h (B) = γ(b h), B B(), also denoted by N (h, Q), is absolutely continuous with respect to γ if and only if h H and, in this case, with the usual notation h = Rĥ, we have, see e.g. [7, Corollary 2.4.3], 1 (9) dγ h (x) = exp {ĥ(x) 2 h 2 H dγ(x). It is also important to notice that if we define for any λ R the measure (10) γ λ (B) = γ(λb), B B() then γ λ γ σ if and only if λ = σ (see for instance [7, Example 2.7.4]). 2.3 Gradient, divergence and Sobolev spaces For a given function f : R and h H, we define and f(x + th) f(x) h f(x) := lim. t 0 t hf(x) = h f(x) f(x)ĥ(x), wherever this makes sense. Here, as usual, h = Rĥ. We shall use the shorter notation j = hj, j = h j. The gradient H f : H of f is defined as H f(x) := j N j f(x)h j. 6

7 Notice that if f(x) = g(π m x) with g C 1 (R m ), then h f(x) = g(π m x) Π m h. The operator h is (up to a change of sign) the adjoint of h with respect to L 2 (, γ), namely (11) φ h fdγ = f hφdγ φ, f FCb 1 (). The divergence operator is defined for Φ : H as HΦ(x) := j N j [Φ(x), h j ] H. We define the space FCb 1 (, H) of cylindrical H-valued functions as the vector space spanned by functions φh, where φ runs in FCb 1 () and h in H. With this notation, the integration by parts formula (11) gives (12) [ H f, Φ] H dγ = f HΦ dγ for every f FCb 1(), Φ FC1 b (, H). Thanks to (12), the gradient H is a closable operator in the topologies L p (, γ), L p (, γ, H) for any p [1, ) and we denote by D 1,p (, γ) the domain of its closure. Notice that the space denoted by D 1,p (, γ) by Fukushima in [18] is denoted by W p,1 (, γ) in [7]. Anyway, these spaces coincide, see [7, Section 5.2] and (12) holds for every f D 1,p (, γ), Φ FCb 1 (, H). Let us recall the Gaussian isoperimetric inequality, see [20]. Let E, and set B r = {x H : x H < r, E r = E + B r ; then (13) Φ 1 (γ(e r )) Φ 1 (γ(e)) + r, with Φ(t) = Then, setting t e s2 /2 2π ds. (14) U (t) = (Φ Φ 1 )(t) t 2 log(1/t), as t 0; the inequality (15) γ(e r ) γ(e) + ru (γ(e)) + o(r) follows. The isoperimetric inequality implies also the following Gauss Sobolev inequality (16) H f L 1 (,γ) 0 U (γ({ f > s)) ds, which implies the continuous embedding of D 1,1 (, γ) into the Orlicz space L log 1/2 L(, γ) := { u : R measurable : A 1/2 (λ u ) L 1 (, γ) for some λ > 0, 7

8 endowed with the Luxenbourg norm u L log 1/2 L := inf {λ > 0 : A 1/2 ( u /λ) dγ 1, see [19, Proposition 3.2]. Here A 1/2 is defined by A 1/2 (t) := t 0 log 1/2 (1 + s)ds. Analogously, using the continuity of the map f f p from D 1,p (, γ) to D 1,1 (, γ), one obtains that D 1,p (, γ) embeds continuously into the Orlicz space (17) L p log 1/2 L(, γ) := { u : R meas. : A 1/2 (λ u p ) L 1 (, γ) for some λ > 0. Finally we recall the Poincaré inequality, see [7, Theorem ]: for any u D 1,p (, γ), p 1 p (18) u udγ dγ C p H u p H dγ, where C p depends only on p. 2.4 The Ornstein Uhlenbeck semigroup Let us define the Ornstein Uhlenbeck semigroup (T t ) t 0, by Mehler s formula ( (19) T t u(x) = u e t x + ) 1 e 2t y dγ(y) for all u L 1 (, γ), t 0. For our purposes, the following properties of the Ornstein Uhlenbeck semigroup are relevant: T t is a contraction semigroup in L 1 (, γ) and T t u D 1,1 (, γ) for any u L log 1/2 L(, γ), t > 0, (see [19, Proposition 3.6]). In addition, a direct consequence of (7) and of Jensen s inequality is (20) T t u Y dγ u Y dγ, for any u L 1 (, γ, Y ), with Y Hilbert (here T t is defined componentwise, namely T t u, y = T t u, y ). Stronger smoothing properties of T t hold for bounded functions; in particular if f L p (, γ) for any p > 1, then T t f D k,q (, γ) for any k N, q > 1 (see [7, Proposition 5.4.8]). Moreover, the following commutation relation holds (again componentwise) for any u D 1,1 (, γ): (21) H T t u = e t T t H u, t > 0. Therefore, we get H T t+s u = H T t (T s u) = e t T t H T s u, t 0, s > 0 8

9 for any u L log 1/2 L(, γ), see [7, Proposition 5.4.8]. As a consequence, we obtain that the limit (possibly infinite) (22) I(u) := lim H T t u H dγ t 0 always exists for u L log 1/2 L(, γ). Indeed, consider the map s H T t+s u H dγ = e t T t H T s u H dγ and observe that H T t u H dγ lim inf s 0 e t lim inf s 0 H T t+s u H dγ = e t lim inf s 0 H T s u H dγ by (20), which obviously implies that the limit exists. It also follows from (21) and (12) that (23) T t f HΦdγ = e t f H(T t Φ)dγ, T t H T s u H dγ for all f L log 1/2 L(, γ), Φ FCb 1 (, H). Indeed, we can assume by a density argument that f D 1,1 (, γ) to get T t f HΦdγ = [ H T t f, Φ] H dγ = e t [ H f, T t Φ] H dγ = e t f HT t Φdγ. Another important consequence of (21) is that if u D 1,1 (, γ) then (24) lim t 0 H T t u H u L 1 (,γ) = 0. Finally, notice that if E m u are the canonical cylindrical approximations of a function u L log 1/2 L(, γ) defined in (8) then (25) H T t E m u H dγ H T t u H dγ, t > 0. To prove (25), let us first notice that, by the rotational invariance of γ m, T t E m u = E m T t u. Indeed, T t E m u(x) = = E m u(e t x + 1 e 2t z)dγ(z) u(e t Π m x + 1 e 2t Π m z + y )dγm(y )dγ(z) m 9

10 and by apply the rotation invariance of Gaussian measures (7) to γm we get E m T t u(x) = T t u(π m x + w )dγm(w ) = = m m m m ( u e t (Π m x + w ) + ) 1 e 2t z dγm(w )dγ(z) m ( u e t Π m x + e t w + 1 e 2t z m + 1 e 2t z ) dγm(w )dγm(z )dγ m (z m ) ( = u e t Π m x + 1 e 2t Π m z + y ) dγm(y )dγ(z). m From the above commutation relation it follows that the vector H T t E m u = H E m T t u coincides with its projection m on H m, since E m u depends only on x m m. Moreover, by Jensen s inequality we have H T t E m u(x) H = E m ( m T t u)(x) H = m T t u(π m x + x )dγm(x ) m m T t u(π m x + x ) H dγm(x ) = E m m T t u(x) H. m Since m T t u(x) H H T t u(x) H we can integrate both sides to get (25). 3 BV functions in infinite dimensions We have collected in the preceding section the tools we need in order to discuss BV functions in the Wiener space setting. The BV (, γ) class can be defined as follows. Definition 3.1 (BV space). Let u L log 1/2 L(, γ). We say that u BV (, γ) if there exists a measure µ M (, H) such that for any φ FCb 1 () we have (26) u(x) j φ(x)dγ(x) = φ(x)dµ j (x) j N, where µ j = [h j, µ] H. In particular, if u = χ E and u BV (, γ), then we say that E has finite perimeter. Equivalently, we may require the existence of measures µ j as in (26) satisfying (27) sup (µ 1,..., µ m ) () <. m Indeed, if µ j = [µ, h j ] H, then the total variation of the R m valued measure (µ 1,..., µ m ) in is less than µ (). Conversely, if (27) holds, then the measure µ := j µ jh j is well defined and belongs to M (, H) (it suffices to consider the densities f i of µ i with respect to the measure σ := sup m (µ 1,..., µ m ) to obtain m 1 f i 2 1 σ-a.e., hence (f i ) l2 1 σ-a.e. and µ = f j h j σ). H 10

11 Remark 3.2. Notice that in the previous definition we have required that the measure µ is defined on the whole of B() and is σ-additive there. Since cylindrical functions generate the Borel σ-algebra the measure µ verifying (26) is unique, and will be denoted D γ u. Using (3) the total variation of D γ u is given by D γ u () = sup { D γ u, Φ ; Φ C b (, H), Φ(x) H 1 x (28) { = sup u HΦdγ; Φ FC 1 b (, H), Φ(x) H 1 x Actually it will be convenient to define also an even smaller space FC 1 c (, H) of vector fields Φ representable as follows: Φ(x) = m φ i ( x 1, x,..., x m, x )h i i=1 with h i = Qx i and φ i Cc 1 (R m ). The space FCc 1 (, H) has an additional technical advantage: the divergence H Φ of Φ above is i iφ i φ i R x i and is a bounded function because R x i is bounded on the support of φ i. By a further approximation, based on the fact that any Φ FCb 1 (, H) is the pointwise limit of a uniformly bounded sequence (Φ n ) FCc 1 (, H), we have also (29) { D γ u () = sup u HΦdγ; Φ FCc 1 (, H), Φ(x) H 1 x. In the case u = χ E, we write P γ (E, ) for the measure D γ χ E and we shall also write P γ (E) for P γ (E, ). Remark 3.3 (About the L log 1/2 L(, γ) assumption). The L log 1/2 L(, γ) membership hypothesis we made on u is necessary to give sense to the integral of the product u j φ: indeed, this term is the sum of the function u j φ (which makes sense for u L 1 (, γ) only) and uφĥj, which makes sense by Orlicz duality if u L log 1/2 L(, γ) and exp(c ĥj 2 ) L 1 (, γ) for some c > 0. Since ĥj = R x j for some x j, by our construction of h j, this exponential integrability property follows by Fernique s theorem (5). Nevertheless, we shall provide different equivalent definitions of BV where this extra integrability property is not needed (as in the finite-dimensional case) but rather derived as a consequence. These equivalent definitions will also show that Definition 3.1 is independent of the choice of the basis (h j ). Let us see an equivalent way of defining the BV class, with partial derivatives along all directions h H; in this case, since ĥ is in L2 (, γ) and not better, we have to assume u L 2 (, γ) to give a sense to the integration by parts formula, even when cylindrical test functions are involved. Proposition 3.4. Let u L 2 (, γ). Then, u BV (, γ) if and only if for every h H there is a real measure µ h such that (30) u(x) hφ(x)dγ(x) = φ(x)dµ h (x) φ FCb 1 (), with h H =1 µ h finite. In this case, D γ u = h H =1 µ h.. 11

12 Proof. If u BV (, γ) then the existence of µ h = [h, D γ u] H for all h H follows from the linearity of the h operator with respect to h, and the boundedness of µ h from the finiteness of D γ u. In particular, h H =1 µ h D γ u (). Conversely, define µ j = µ hj and µ = µ j h j, j=1 so that µ h = [h, µ] H. The integration by parts (26) clearly holds; we have to prove that µ is a finite measure. If we fix a partition (B n ) n N of, if µ(b n ) 0, we define so that we obtain α n = µ(b n) µ(b n ) H µ(b n ) H = [µ(b n ), α n ] H α n H = n N n N n N µ αn (B n ) µ h, n N h H =1 and then u BV (, γ) with D γ u := µ and µ h H =1 µ h. µ αn (B n ) It is easy to verify that if u D 1,1 (, γ), then u BV (, γ) with D γ u = H uγ. Now we relate BV functions in R m with cylindrical functions in. We denote as before by γ m the standard Gaussian distribution on R m and we point out that all directional derivatives ν and their adjoints ν have the same meaning as in the infinite dimensional case, but without restriction on directions, since H = in this case; we shall try to use as much as possible a consistent notation, valid both for the finite-dimensional and the infinite-dimensional case. Proposition 3.5. Let u L log 1/2 L(, γ) be a cylindrical function, u(x) = v(π x 1,...,x m x), with R x i orthonormal. Then v BV (Rm, γ m ) if and only if u BV (, γ) and D γ u () = D γm v (R m ). Proof. Recalling that the law of Π under γ is γ m, we have { D γ u () = sup u(x) Φ(x)dγ(x) : Φ FCb 1 (, H), Φ(x) H 1 x { = sup u(x) Φ(x)dγ(x) : Φ FCb 1 (, H m ), Φ(x) H 1 x = sup { v(y) Ψ(y)dγ m (y) : Ψ [Cb 1 (R m )] m, Ψ 1 R m. As a particular example of Sobolev functions we can consider Lipschitz functions. 12

13 Definition 3.6 (H-Lipschitz functions). A Borel function f : R is said to be H- Lipschitz if there exists a constant C such that for γ-a.e. x one has (31) f(x + h) f(x) C h H, h H. It can be proved that for a H-Lipschitz function f there exists a full-measure γ-measurable set 0 such that 0 + H = 0 and, for every x 0, one has f(x + h) f(x) C h H, h H. In particular, f has a version such that the previous inequality is satisfied for every x. By the arguments in [7, Section 5.11], it can be proved that H-Lipschitz functions belong to D 1,p (, γ) for every p 1, and in particular to BV (, γ). An important result is the following coarea formula, which can be proved by following verbatim the proof of [15, Section 5.5]. Theorem 3.7. If u BV (, γ), then for every Borel set B the following equality holds: (32) D γ u (B) = P γ ({u > t, B)dt. R As a corollary, we have that almost every ball has finite perimeter, since the distance function is H Lipschitz. We do not know whether every ball has finite perimeter, because P γ (B r (x)) is not trivially monotone with respect to r and no homothety argument can be used in view of (10). We now extend from finite dimensions to infinite dimensions some typical tools of the BV theory. The following definition is a very convenient tool in the theory of BV functions: Definition 3.8 (Total variation). We define total variation of a function v L 1 (, γ) by { (33) V γ (v) := sup v HΦdγ : Φ FCc 1 (, H), Φ(x) H 1 x. The name is justified by the following observation: if v BV (, γ), then (29) shows that V γ (v) = D γ v (); on the other hand, if is finite-dimensional and the supremum in (33) is finite, then a direct application of Riesz theorem provides us with a -valued measure µ, with total variation less than V γ (v), such that v HΦdγ = Φ, µ Φ Cb 1 (, H). Hence µ = D γ u and V γ (u) = D γ v (). This equivalence is much less obvious in the infinite-dimensional case, since Riesz theorem is not available, and it will be discussed in the next section. Notice that v V γ (v) is lower semicontinuous with respect to the L 1 (, γ) convergence, since it is the supremum of a family of continuous functionals. Since v V γ (v) is easily seen to be nonincreasing under cylindrical approximation (with the same argument used in Proposition 3.5), we can combine this property with lower semicontinuity to get (34) V γ (v) = lim m V γ(e m v). Analogous definitions can be given for the total variation along a direction ν H m for some m; in this case, in order to have a bounded adjoint derivative νφ, we consider the space F ν C 1 c () of cylindrical functions φ with support contained in a strip {a < x, x < b, where ν = Qx. 13

14 Definition 3.9 (Directional total variation). Let ν m H m be a unit vector. We define total variation of a function v L 1 (, γ) along ν by { (35) Vγ ν (v) := sup v νφdγ : φ F ν Cc 1 (), φ(x) H 1 x. Again v Vγ ν (v) is lower semicontinuous with respect to the L 1 (, γ) convergence and, in finite dimensional spaces, Riesz theorem shows that the quantity is finite if and only if the integration by parts formula (36) v νφdγ = φdµ φ Cb 1 () holds for some real-valued measure µ with finite total variation, that we shall denote by D ν γv; if this happens, µ () coincides with V ν γ (v). Finally, (37) Vγ ν (v) = lim V γ ν (E m v). m We can now discuss 1-dimensional sections of Gaussian BV functions in the same spirit as Section 3.11 of [1]. Notice that any u BV (R m, γ m ) is in the classical space BV loc (R m ), so that we can use all the (local) properties known in Euclidean case, and (38) D γ u = G m Du where G m (x) = (2π) m/2 exp( x 2 /2) is the standard Gaussian kernel and Du stands for the classical derivative of u in the sense of distributions. Let us fix a unit direction ν = Qx H, let Π(x) = x, x be the induced projection and let us write x as y + Π(x)ν. Then, denoting by K the kernel of Π, γ admits a product decomposition γ = γ γ 1 with γ Gaussian in K. For u : R and y K we define the function u y : R R by u y (t) = u(y + tν). Theorem Let u L log 1/2 L(, γ) and let ν m H m ; then Vγ ν (u) = V γ1 (u y ) dγ (y). K In particular Definition 3.9 is independent of the choice of the basis and makes sense for all h H. Proof. Let us fix φ F ν Cb 1() with φ 1. Then ( ) u(x) νφ(x)dγ(x) = u y (t)(φ y(t) tφ y (t)dγ 1 (t) dγ (y) whence K R Vγ ν (u) V γ1 (u y )dγ (y). K K V γ1 (u y )dγ (y), For the reverse inequality we can assume that V ν γ (u) is finite. First we prove the inequality in the finite-dimensional case = R m and γ = γ m, and then we consider the general case. If = R m and γ = γ m then the measure µ = D ν γ m u in (36) is a real valued measure 14

15 with total variation in R m less than Vγ ν (u). Let us show that we may find a sequence (u n ) C (R m ) D 1,1 (R m, γ m ) such that u n u in L 1 (R m, γ m ) and lim ν u n (y) dγ m (y) Vγ ν n m (u). R m For, set u n = T 1/n u and notice that for every φ F ν Cb 1(Rm ) with φ 1 φ ν u n dγ m = νφu n dγ m νφudγ m Vγ ν m (u), R m R m R m then the sequence ( ν u n ) is bounded in M (R m ), and (up to a subsequence which we do not relabel) weakly converges to a measure µ. The above limit relation shows that µ = Dγ ν m and that the whole sequence is convergent. By Fubini theorem (possibly passing to a subsequence) for γ -a.e. y K the sequence (u n,y ) converges to u y in L 1 (R, γ 1 ) and then by semicontinuity we get Vγ ν (u) lim inf ν u n (y) dγ(y) = lim inf u n,y(t) dγ 1 (t)dγ (y) n + R m n + K R = lim inf V γ1 (u n,y )dγ (y) V γ1 (u y )dγ (y). n + K In the infinite-dimensional case we consider the cylindrical approximations v m := E m u; since they converge in L 1 (, γ) we can find a subsequence (m i ) such that v mi,y u y in L 1 (R, γ 1 ) for γ -a.e. y K; then, lower semicontinuity of v V γ1 (v), Fatou s lemma and monotonicity give K V γ1 (u y )dγ (y) lim inf i K K V γ1 (u miy)dγ (y) sup V γ (v m ) V γ (u). m Corollary Let h = Rĥ H and c R; then the sets E = {x : ĥ(x) c have finite perimeter with (39) P γ (E) = 1 exp { c2 2π 2 h 2. H Proof. With no loss of generality we can assume that h H = 1. We prove the identity first in the case when ĥ = R x for some x ; without loss of generality, we may assume that Qx H = 1. In this case the assertion simply follows by noticing that E is a cylindrical set of the form { E = x : x, x B, with B = {s R : s c. This implies that P γ (E) = P γ1 (B) = e c2 /2 / 2π. In the general case, density of Q in H and lower semicontinuity of the perimeter provides the inequality in (39); to prove the inequality we fix φ Cc (R) with φ(c) = 1, φ 1, k = Qk, k H = 1 and k h 2 H < 2ε and the field Φ ε(x) = φ( x, k )k; then P γ (E) kφ( x, k )dγ. E 15

16 By Proposition 2.2 and considering the map x (ĥ(x), x, k ), the right hand side can be represented as {x 1 c φ(x 2 )dη ε (x 1, x 2 ), where η ε are Gaussian in R 2 with γ 1 as marginals and x 1 x 2 dη ε (x 1, x 2 ) > 1 ε; as ε 0 these Gaussians converge to the standard Gaussian on the diagonal of R 2, so that P γ (E) {z c φ(z)dγ 1 (z) = φ(c) 2π e c2 /2. In connection with the proof of the previous Corollary, notice that it would be desirable to extend Theorem 3.10 even to the case when ν = Rĥ H; however, this extension presents some difficulties, since ĥ is not really a linear map on. 4 Main results We are now in a position to characterize BV functions in the same way as they are characterized in the classical case. Notice that in the classical case the original definition of BV given by E. De Giorgi in [12] was based on property (4) below, with the heat semigroup instead of the Ornstein-Uhlenbeck one. Theorem 4.1. Given u L 1 (, γ), the following are equivalent: (1) u belongs to BV (, γ); (2) the quantity V γ (u) in (33) is finite; { (3) L γ (u) := inf lim inf n Hu n H dγ : u n D 1,1 L (, γ), u 1 n u < ; (4) u L log 1/2 L(, γ) and the quantity I(u) in (22) is finite. Moreover, D γ u () = V γ (u) = L γ (u) = I[u] and (40) H T t u H dγ e t D γ u () t > 0. Proof. (1) (2). Simply comparing the classes of competitors, we notice that V γ (u) D γ u (). (2) (3). Let t n 0 and u n = T tn u. Then, for all Φ FCb 1(, H) with Φ(x) H 1 for every x, from (23) we deduce [ H u n, Φ] H dγ = e tn u (T tn Φ)dγ V γ (u). Therefore, H u n L 1 (,γ) V γ (u). In particular, we have proved that L γ (u) V γ (u). (3) (4). Let (u n ) n N be such that u n u in L 1 (, γ) and H u n L 1 (,γ) L γ (u). Then, H T t u H dγ lim inf n H T t u n H dγ = e t lim inf H u n H dγ = L γ (u). n 16

17 In addition, Fatou s lemma and (16) yield (41) 0 U (γ({ f > s)) ds L γ (u) <, so that u L log 1/2 L(, γ). Observe that in particular we have proved that I(u) L γ (u). (4) (1). We first prove that for all j the derivative µ j along the direction h j exists, and then we prove (27) to conclude that u BV (, γ). Since T t u D 1,1 (, γ) for t > 0, we have Vγ hj (T t u) = Dγ hj T t u = j T t u dγ. In particular, setting ν = h j and adopting the same notation as in Theorem 3.10, we have V γ1 ((T t u) y )dγ (y) = j T t u dγ H T t u H dγ. K Now we can find t n 0 sufficiently fast in such a way that (T tn u) y converge to u y in L 1 (R, γ 1 ) for γ -a.e. y K and conclude, by the lower semicontinuity of v V γ1 (v), that V γ1 (u y )dγ (y) I(u) <. K It follows that for γ -a.e. y K the function u y has bounded variation in R. By a Fubini argument, based on the factorization γ = γ γ 1, the 1-dimensional integration by parts formula yields that the measure µ j = D γ1 u y γ, i.e. µ j (A) = D γ1 u y (A y )dγ (y) K (where A y := {t : y + th j A is the y-section of a Borel set A) provides the derivative of u along h j. Notice that µ j is well defined, since we have just proved that K D γ 1 u y (R)dγ is finite. Now, setting µ i = Dγ hi u, we check (27); by a density argument, it suffices to prove that m φ i dµ i I(u) i=1 for all φ i FCb 1() with i φ2 i 1; by integration by parts, it suffices to show that m lim sup φ i ddγ hi T t u I(u) t 0 or equivalently lim sup t 0 i=1 i=1 m φ i i T t udγ I(u). The latter inequality is trivial, since m 1 φ i hi T t u H T t u H. Passing to the limit as s 0 in the inequality H T t+s u H dγ e t H T s u H dγ provides HT t u H dγ e t I(u) and concludes the proof. 17

18 Remark 4.2. Arguing as in the proof of the implication (2) (3) we see that for all u BV (, γ) and all closed subspaces K H the following inequality holds: lim sup π K H T t H dγ Dγ K u (). t 0 Here π K : H K is the orthogonal projection and D K γ u = π K D γ u. Remark 4.3. It is worth noticing that sets of finite perimeter E can also be characterized using the functional J E (t) = U (T t χ E ) 2 + H T t χ E 2 H dγ, in the spirit of [5]. In fact, it can be proved that (42) d dt J E(t) 0, lim J E (t) = P γ (E). t 0 With minor modifications the proof of Theorem 4.1 allows also to show that, for all j, the family of measures j T t uγ is tight and the limit as t 0 is D γ u, h j, in the duality with C b (); in addition, the limit of D γ u, h j γ is D γ u, h j. We give just a sketch of proof, since this result is a consequence, rather than a tool, in our characterization theorem of BV functions. We also notice that similar results could be stated and proved for the measures π K H T t u H γ and π K D γ u, with K H closed subspace. Theorem 4.4. Let u BV (, γ) and j 1. Then lim j T t uγ = D γ u, h j, t 0 lim j T t u γ = D γ u, h j t 0 in the duality with C b (). Proof. Since, by the integration by parts formula, j T t uγ weakly converge to D γ u, h j in the duality with FCb 1() as t 0, in order to show the convergence of jt t uγ it suffices to show that j T t u γ is tight. Indeed, this ensures by Prokhorov theorem the compactness in the duality with C b (), and the weak limit must be the same as above, by the density of FCb 1() in C b(). By Remark 4.2 we have lim sup j T t u dγ D γ u, h j (), t 0 hence tightness is achieved, thanks to Lemma 2.1, by lim inf j T t u dγ D γ u, h j (A) for all A open. t 0 A This inequality, in turn, can be derived as in the proof of the implication (3) (4) in the proof of Theorem 4.1, using the fact that the sections A y = {t R : y + th j A are open, and the lower semicontinuity of v BV (R, γ 1 ) D γ1 v (J) with respect to the L 1 (γ 1 ) convergence, for all J R open. 18

19 An immediate consequence of Theorem 4.1 is the extension, by approximation, of many properties of Sobolev functions to BV : for instance BV (, γ) L (, γ) is an algebra (and therefore sets of finite perimeter are stable under union and intersection), BV (, γ) is stable under left composition with Lipschitz maps f, and D γ f u () Lip(f) D γ u (), etc. We need in particular the inequalities stated in the following proposition, a direct consequence of (16) and (18) for D 1,1 (, γ) functions, and of the equality D γ u () = L γ (u). Proposition 4.5 (Sobolev and Poincaré inequalities). Let u BV (, γ). Then and + 0 U (γ({ u > s))ds D γ u () u udγ dγ C 1 D γ u (), where C 1 is the constant in the Poincaré inequality (18) for p = 1. The following approximation result for sets of finite perimeter is a consequence of the approximation in BV through smooth functions and the coarea formula. Proposition 4.6. Let E be a set with finite perimeter; then there exist cylindrical sets E j = Π 1 m j B j, with B j R mj smooth sets, such that lim χ E j χ E L j 1 (,γ) = 0 and lim P γ(e j ) = P γ (E). j Proof. According to Theorem 4.1, for every u BV (, γ) there is a sequence of D 1,1 (, γ) functions such that the L 1 norms of their gradients converge to the total variation of u. Moreover, smooth cylindrical functions are dense in D 1,1 (, γ), hence there exists a sequence (u j ) of smooth cylindrical functions with u j χ E in L 1 (, γ) and H u j H dγ P γ (E). Assuming with no loss of generality that 0 u j 1, the conclusion then follows from the coarea formula by taking smooth levels B j of u j. Due to the previous proposition, we say that E is a smooth set if E = Π 1 m B for some set B R m with smooth boundary. Denoting by H m 1 the Hausdorff (m 1)-dimensional measure in R m, since by (38) D γm χ B = G m Dχ B and Dχ B (A) = H m 1 (A B) for all Borel sets A, for the sets E j of the previous proposition we get P γ (E) = lim j + B j G mj dh mj 1. Remark 4.7. Using the results in [5], it is possible to repeat the same argument contained in [8] to show that the isoperimetric inequality and extremality of half spaces hold in infinite dimension as well. That is, for any E with finite perimeter, P γ (E) U (γ(e)), with U defined in (14) with equality if and only if E is an arbitrary half space. The proof is based on analysis of the quantity J E introduced in Remark 4.3; in particular, once one has (42), by using the fact that J E (t) is twice differentiable, (see [7, Proposition 5.4.8]), one shows that if equality holds for a set E in the isoperimetric inequality, then χ E is affine (see [8, Lemma 2.1]), that is E is a half space. 19

20 In the next corollary we consider a finite dimensional subspace K H. We shall denote by Π K : K the canonical projection (induced by the choice of a basis of K) and set Π K (x) = x Π K(x); since Π K Π K = Π K and Π K K = Π K we may write in a unique way x = x 1 + x 2 with x 1 K and x 2 Ker(Π K ) and, accordingly, u x1 (x 2 ) = u(x 1 + x 2 ). Setting 1 = K and 2 = Ker(Π K ) (the closure of K in ), we have also the factorization dγ(x 1, x 2 ) = dγ 1 (x 1 ) dγ 2 (x 2 ) with γ 1 Gaussian in 1 and γ 2 Gaussian in 2 ; furthermore, the Cameron-Martin spaces are respectively K and K. The next proposition is the natural complement of Theorem 3.10: in that theorem the slices are 1-dimensional, and with minor changes the same result could be proved for finitedimensional slices. Here we consider, instead, slices of finite codimension; for the sake of simplicity we state it assuming a priori that the map u is globally BV, hence without using variations, and we consider only one implication. In the sequel, for a given closed subspace L H and u BV (, γ) we set, in accordance with Remark 4.2, D L γ u := π L D γ u M (, L), where π L : H L is the orthogonal projection. Proposition 4.8. Let u BV (, γ); then u x1 BV ( 2, γ 2 ) for γ 1 -a.e. x 1 1 and (43) Dγ K u () = D γ2 u x1 ( 2 )dγ 1 (x 1 ). 1 Proof. Let u n = T tn u, with t n 0, and assume with no loss of generality that (u n ) x1 converge to u x1 in L 1 ( 2, γ 2 ) for γ 1 -a.e. x 1. We have K (u n ) x1 (x 2 ) K dγ 2 (x 2 )dγ 1 (x 1 ) = K u n H dγ 2 1 and, passing to the limit as n, Fatou s lemma and Remark 4.2 give L γ2 (u x1 )dγ 1 (x 1 ) D K γ u (), 1 with L γ2 as in (3) of Theorem 4.1. From Theorem 4.1 we deduce u x1 BV ( 2, γ 2 ) for γ 1 -a.e. x 1 1 and the inequality holds in (43). Arguing as in the first part of the proof of Theorem 3.10 we see that the factorization γ = γ 1 γ 2 yields h, D γ u = h, D γ2 u x1 dγ 1 (x 1 ) 1 for all h K (indeed, both measures satisfy the integration by parts formula in the direction h), hence Dγ K u = D γ2 u x1 dγ 1 (x 1 ). 1 This immediately gives Dγ K u () D γ2 u x1 ( 2 )dγ 1 (x 1 ). 1 20

21 Corollary 4.9. Let u BV (, γ) let K H be finite-dimensional and E K u the conditional expectation relative to K. If K is the complementary subspace of K, we have u E K u dγ C 1 Dγ K u (). Proof. We write as in Proposition 4.8 x = x 1 + x 2 with x 1 1 = K and x 2 2 = Ker(Π K ), and denote by u x1 (x 2 ) = u(x 1 + x 2 ); using the Poincaré inequality in K we get u(x) E K u(x) dγ(x) = u(x 1 + x 2 ) E K u(x 1 ) dγ 1 (x 1 )dγ 2 (x 2 ) 1 2 = u x 1 (x 2 ) u x1 (z)dγ 2 (z) dγ 2(x 2 )dγ 1 (x 1 ) C 1 D K γ 2 u x1 ( 2 )dγ 1 (x 1 ) = C 1 Dγ K u (), 1 where in the last line we have used Proposition 4.8. In an analogous way one can prove that (44) u E K u p dγ C p π K H u p H dγ u D1,p (, γ). The next Theorem generalises Theorem of [7], for the part concerning Sobolev spaces. The part concerning BV functions is, to our knowledge, totally new. Theorem The following statements hold. (i) For p > 1, let F be a bounded family of functions in D 1,p (, γ); assume that for every ε > 0 there exists a finite dimensional subspace K of H such that, (45) K u p Hdγ ε, u F; then F is relatively compact in L p (, γ). (ii) Let F be a bounded family of functions in BV (, γ); assume that for every ε > 0 there exists a finite dimensional subspace K of H such that, (46) D K u () ε, u F; then F is relatively compact in L 1 (, γ). Proof. We first discuss briefly the finite-dimensional case, = R m, γ = γ m, for BV functions (for Sobolev functions, see [6, Theorem ] or adapt the argument below, taking into account the continuous embedding of D 1,p into the space L p log 1/2 L in (17)). Since the family F is bounded also in L log 1/2 L(R m, γ m ), we obtain that F is equi-integrable in L 1 (R m, γ m ), hence by a truncation argument we can assume with no loss of generality that F is uniformly bounded also in L (R m, γ m ). Under this assumption relative compactness follows obviously from relative compactness in L 1 loc (Rm ); the latter is a consequence of the classical compact embedding of BV loc in L 1 loc and of the identity D γ m u = G m Du, showing that sup u F Du (K) is finite for any compact set K R m. 21

22 We shall prove only (ii) by showing that F is totally bounded (the proof of (i) is analogous). By Corollary 4.9 we have u E K u dγ C 1 ε u F. Since the result holds in finite dimension, the family F K = {E K u : u F is totally bounded in L 1 (, γ) and the thesis follows. Remark Statement (i) is not true in D 1,1 (, γ) under condition (45) with p = 1. In this case the family F is only pre compact and the limit is in general only in BV (, γ). Moreover, bounded families in BV (, γ) are not in general pre compact; as an example it suffices to consider the family F of characteristic functions of the sets {x : x, x 0, x, Qx H = 1. Condition (45) is satisfied if there exist a compact operator K on H such that H u K(H) almost everywhere for all u F and K 1 H u p Hdγ is uniformly bounded on F. Indeed, a simple compactness argument proves that h Π m h H 0 uniformly on {h K(H) : K 1 h H 1, hence we may find ω m 0 such that H u Hm u p H dγ ω m K 1 H u p H dγ. Analogously, in the BV case we have that (46) is fulfilled if there exists a compact operator K on H such that the Radon-Nikodym densities D γ u/ D γ u belong to K(H) D γ u -a.e. for all u F and D γu K 1 D γ u d D γ u H is uniformly bounded on F. Indeed, D H m γ u () = D γ u Dγ Hm u () = D γ u D γ u π D γ u H m D γ u d D γ u H ω m D γu K 1 D γ u d D γ u. H H 5 The case when is a Hilbert space The results presented above show that there are strict links between the notion of derivative h, the semigroup T t and the measure γ, which turns out to be invariant under T t. Indeed, if is an Hilbert space, different notions of derivative can be given, related to different semigroups still having γ as invariant measure (see Remark 5.2). In this section we briefly describe another point of view and confine ourselves to deriving the corresponding compact embedding theorem both for Sobolev and BV functions. In this section we assume that (,, ) is a separable Hilbert space; let γ = N (0, Q) be as before. Identifying and its dual with the inner product, we fix an orthonormal basis (e k ) in such that Qe k = λ k e k, k 1, 22

23 with λ k > 0 and k λ k <. If we set x k = x, e k, since R e k = x k and Rx k = λ k e k it follows that λ k e k H = x k L 2 (x,γ) = λ k. Consequently an orthonormal basis in H is given by ε k = λ 1/2 k e k, the Cameron-Martin norm is ( k x2 k /λ k) 1/2 and H = Q 1/2 = { { x : y with x = Q 1/2 y = x : Notice that Q 1/2 is still a compact operator on. Setting for all k 1 f(x + te k ) f(x) D k f(x) = ek f(x) = lim, t 0 t k=1 x 2 k <. λ k we define by linearity a gradient operator D : FCb 1() FC b(, ). The gradient turns out to be a closable operator with respect to the topologies L p (, γ) and L p (, γ, ) for every p 1, and we denote by W 1,p (, γ) the domain of the closure in L p (, γ), endowed with the norm ( ( u 1,p = u(x) p dγ + D k u(x) 2) p/2 ) 1/p, dγ where we keep the notation D k also for the closure of the partial derivative operator. As a consequence of the relation ε k = λ 1/2 k e k we have also k=1 (47) εk = λ 1/2 k D k, so that W 1,p (, γ) D 1,p (, γ), since H f H = ( k λ k D k f 2 ) 1/2. Since ê k = x k /λ k, the integration by parts formula (11) becomes (48) g(x)d k f(x)dγ(x) = f(x)d k g(x)dγ(x) + 1 x k f(x)g(x)dγ(x) λ k k for f, g FCb 1(). However, we point out that even though D vf = i v id i f makes sense for v, the corresponding integration by parts along v does not, since at least convergence of v k x k dγ = 1 v k λ k 2π λk is needed (this is ensured for v Q() = Q 1/2 (H)). In this context, we may give a corresponding definition of BV functions. k Definition 5.1. A function u L 1 (, γ) belongs to BV (, γ) if there exists ν u M (, ) such that for any k N we have u(x)d k ϕ(x)dγ = ϕ(x)dνk u + 1 x k u(x)ϕ(x)dγ, ϕ FCb 1 (), λ k with ν u k = νu, k. 23

24 It is immediate to check that BV (, γ) is contained in BV (, γ) and that (49) D γ u, ε i = λ 1/2 i ν u i. Remark 5.2. Even though we do not further pursue this point of view here, let us point out that the transition semigroup corresponding to the gradient D is given by R t f(x) = f(e tq 1 /2 x + y)dγ t (y), f C b (), where γ t = N (0, Q t ) and Q t = Q t 0 e sq 1 ds = Q(1 e tq 1 ). Notice that e tq 1 e ωt, t 0, where ω = inf 1 λ k. Therefore N (0, Q t ) N (0, Q) = γ weakly as t, so that γ is invariant for R t. Moreover R t maps L 1 (, γ) into W 1,1 (, γ) for every t > 0, see e.g. [11, Proposition ], and this is coherent with the hypothesis u L 1 (, γ) in Definition 5.1. In [3] we plan to investigate further relations between BV (, γ) and the semigroup R t. Let us show that both Sobolev and BV spaces in the present context are compactly embedded into the corresponding Lebesgue spaces. The following statement easily follows from Theorem 4.10 and the relation (47). Theorem 5.3. For every p 1, the embedding of W 1,p (, γ) into L p (, γ) is compact. The embedding of BV (, γ) into L 1 (, γ) is compact. Proof. Let us prove the statement in the Sobolev case, that of BV is similar and uses (49). It suffices to show that every family F bounded in W 1,p (, γ) is totally bounded in L p (, γ). If u 1,p C for all u F, then in particular whence by (47) ( D k u 2) p/2 dγ C p k=1 ( 1 εk u 2 ) p/2dγ C p λ k k=1 By applying Remark 4.11 with K = Q 1/2 the thesis follows. References u F, u F. [1] L. Ambrosio, N. Fusco, D. Pallara: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, [2] L. Ambrosio, S. Maniglia, M. Miranda Jr, D. Pallara: Towards a theory of BV functions in abstract Wiener spaces, Evolution Equations: a special issue of Physica D forthcoming. 24

25 [3] L. Ambrosio, G. Da Prato, D. Pallara: In preparation. [4] L. Ambrosio, M. Miranda Jr, D. Pallara: Special functions of bounded variation in doubling metric measure spaces, in: D. Pallara (ed.), Calculus of Variations: Topics from the Mathematical Heritage of Ennio De Giorgi, Quaderni di Matematica, vol. 14 (2004), Dipartimento di Matematica della seconda Università di Napoli, [5] D. Bakry, M. Ledoux: Lévy Gromov s isoperimetric inequality for an infinite dimensional diffusion generator, Invent. Math., 123 (1996), [6] M. Bertoldi, L. Lorenzi: Analytical methods for Markov semigroups. Chapman and Hall, [7] V. I. Bogachev: Gaussian Measures. American Mathematical Society, [8] E.A.Carlen, C.Kerce: On the cases of equality in Bobkov s inequality and Gaussian rearrangement, Calc. Var. Partial Differential Equations, 13 (2001), [9] G. Da Prato: An introduction to infinite-dimensional analysis, Springer, [10] G. Da Prato, P. Malliavin, D. Nualart: Compact families of Wiener functionals, C. R. Acad. Sci. Paris Sér. I Math., 315 (1992), [11] G. Da Prato, J. Zabczyk: Second order partial differential equations in Hilbert spaces, London Mathematical Society Lecture Note Series, 293, Cambridge U. P., [12] E. De Giorgi: Su una teoria generale della misura (r 1)-dimensionale in uno spazio ad r dimensioni, Ann. Mat. Pura Appl. (4) 36 (1954), , and also Ennio De Giorgi: Selected Papers, (L. Ambrosio, G. Dal Maso, M. Forti, M. Miranda, S. Spagnolo eds.) Springer, 2006, English translation, Ibid., [13] E. De Giorgi: Su alcune generalizzazioni della nozione di perimetro, in: Equazioni differenziali e calcolo delle variazioni (Pisa, 1992), G. Buttazzo, A. Marino, M.V.K. Murthy eds, Quaderni U.M.I. 39, Pitagora, 1995, [14] A. Ehrhard: Symétrisation dans l espace de Gauss, Math. Scand., (2) 53 (1983), [15] L.C. Evans, R.F. Gariepy: Lecture notes on measure theory and fine properties of functions, CRC Press, [16] D. Feyel, A. de la Pradelle: Hausdorff measures on the Wiener space, Potential Anal. 1 (1992), [17] M. Fukushima: On semimartingale characterization of functionals of symmetric Markov processes, Electron J. Probab., 4 (1999), [18] M. Fukushima: BV functions and distorted Ornstein-Uhlenbeck processes over the abstract Wiener space, J. Funct. Anal., 174 (2000), [19] M. Fukushima & M. Hino: On the space of BV functions and a Related Stochastic Calculus in Infinite Dimensions, J. Funct. Anal., 183 (2001),