Fonctions on bounded variations in Hilbert spaces

Transcription

1 Fonctions on bounded variations in ilbert spaces Newton Institute, March 31, 2010

2 Introduction We recall that a function u : R n R is said to be of bounded variation (BV) if there exists an n-dimensional vector measure Du with finite total variation such that u(x)divf(x)dx = F(x), Du(dx), F C0 1 (; ). The set of all BV functions is denoted by BV (R n ). Moreover, the following result holds, see E. De Giorgi, Ann. Mar. Pura Appl

3 Theorem 1 Let u L 1 (R n ). Then (i) (ii). (i) u BV (R n ). (ii) We have where T t is the heat semigroup lim DT t u(x) dx <, (1) t 0 T t u(x) := 1 e 4πt R x y 2 4t u(y)dy. n

4 As well known functions from BV (R n ) arise in many mathematical problems as for instance: finite perimeter sets, surface integrals, variational problems in different models in elasto plasticity, image segmentation, and so on. Recently there is also an increasing interest in studying BV functions in general Banach or ilbert spaces in order to extend the concepts above in an infinite dimensional situation.

5 A definition of BV function in an abstract Wiener space, using a Gaussian measure µ and the corresponding Dirichlet form, has been given by M. Fukushima, JFA 2000 and M. Fukushima, M. ino, JFA 2001.

6 A more analytic different approach was presented in L. Ambrosio, M. Miranda, S. Maniglia and D. Pallara, Phisica D (to appear) and JFA, The definition of BV functions in both approches is based on the Malliavin Sobolev space D 1,1 (, µ) where is the Cameron Martin space and where the corresponding Mehler semigroup replaces the heat semigroup of De Giorgi s theorem.

7 In this talk I shall present a different definition of BV function, following the paper Ambrosio, DP, Pallara preprint We consider a nondegenerate Gaussian measure µ in a seperable ilbert space rather than in a Wiener space. Moreover, our definition of BV function will involve the Sobolev space W 1,1 (, µ) instead of D 1,1 (, µ) as in the previous papers.

8 We give a characterization of BV functions, which generalizes the De Giorgi theorem quoted before, in terms of an Ornstein Uhlenbeck semigroup having µ as invariant measure. It is well known that there are infinitely many such a semigroups. We shall choose the one which is strong Feller, unlike the Melher semigroup, and whose generator is elliptic.

9 Plan of the talk 1 Basic notations and prerequisites including definition of the Sobolev space W 1,1 (, µ) and the O. U. semigroup R t. 2 Definition of BV functions. 3 Generalization of De Giorgi s theorem. 4 The case of a non Gaussian measures. Points 2 and 3 concern the joint paper with L. Ambrosio and D. Pallara, preprint In 4 we shall present some result of a work in progress with B. Goldys.

10 1 Basic notations and prerequisites separable ilbert space. µ non degenerate Gaussian measure of mean 0 and covariance Q. (e k ) complete orthonormal system in and (λ k ) sequence of positive numbers such that We set x k = x, e k, k N. Qe k = λ k e k, k N. We denote by A the self ajoint operator 1 2 Q 1. Then where α k = 1 2λ k. Ae k = α k e k, k N,

11 Integration by parts formula For all k N the following identity is well known, D k u(x) ϕ(x) µ(dx) = u(x)dk ϕ(x) µ(dx), u, ϕ F C1 b (), (2) where F Cb 1() is set of all C1 functions depending only on a finite number of x k which are bounded with their derivatives and D k is given by D is the adjoint of D k in L 2 (, µ). k ϕ = D kϕ + x k λ k ϕ, (3)

12 By (2) it follows that the gradient operator D : F C 1 b () L1 (, µ) L 1 (, µ; ), ϕ Dϕ is closable, see B. Goldys, F. Gozzi and J. Van Neerven, Pot. An We shall denote by W 1,1 (, µ) the domain of the closure of D in L 1 (, µ).

13 The Ornstein Uhlenbeck semigroup We denote by R t the Ornstein Uhlenbeck semigroup R t ϕ(x) = ϕ(e ta x + y)µ t (dy), (4) where µ t is the Gaussian measure of mean 0 and covariance Q t := Q(1 e 2tA ). It is well known that R t is symmetric and that µ is its unique invariant measure. Moreover for any t > 0 and any ϕ B b () one has R t ϕ C b ().

14 2 Functions of bounded variation Let us first recall the definition of vector measure. A -valued measure ζ in (, B()) is a countably additive mapping ζ : B(), A ζ(a). The total variation of ζ is the real countably additive measure on (, B()) defined by { } ζ (K ) := sup ζ(f k ) : (F k ) D(F), k=1 where D(F) is the set of all disjoint decompositions of the Borel set F. We say that ζ has bounded total variation if the measure ζ is finite.

15 By M (, ) we mean the set of all -valued measures with bounded total variation. Let ζ M (, ). For any h N we set ζ h (I) = ζ(i), e h, I B(). Then ζ h is a finite measure in (, B()) and we have ζ(a) = ζ h (A)e h, h=1 A B().

16 Definition Let u L 1 (, µ). We say that u is of bounded variation (u BV (, µ)) if there exists Du M (, ) such that u(x) Dh ϕ(x) µ(dx) = ϕ(x)d (D h u)(dx), h N, ϕ F C 1 b (). (5) where (D h u)(i) = (Du)(I), e h, I B().

17 A criterion to check bounded variation For any u L 1 (, µ) let us consider R(u) := sup m { m k=1 ud k ϕ kdµ : ϕ k C 1 b (), m i=1 ϕ 2 k (x) 1 } (6) It is clear that if u BV (, µ) we have R(u) Du ()

18 A converse result holds Proposition 2 Let u L 1 (, µ). If R(u) < then u BV (, µ) and Du () R(u).

19 Sketch of Proof Assume that R(u) < and m N. Then by (6) there exists a R n -valued measure (D 1 u,..., D m u) such that (D 1 u,..., D m u) R(u). Now, a simple argument shows that u BV (, µ) and Du(I) = (D k u)(i)e k, I B(). k=1

20 3 The main result Theorem 3 Let u L 1 (, µ). Then (i) (ii). (i) u BV (, µ). (ii) For all t > 0 we have R t u W 1,1 (, µ), e ta DR t u L 1 (, µ) (7) and lim inf e ta DR t u dµ <. (8) t 0

21 Remark One can also show that if u BV (, µ) we have DR t u = e ta R tdu, (9) and e ta DR t u dµ Du () (10) lim e ta DR t u dµ = Du (). (11) t 0

22 Basic ingredients of the proof of Theorem 3 The first ingredient is an elementary formula for the commutator between R t and D k. Since D k R t ϕ(x) = e α k t D k ϕ(e ta x + y)µ t (dy), x, we have D k R t ϕ = e α k t R t D k ϕ, ϕ Cb 1 (). (12) Since R t is symmetric we deduce by duality the identity R t D k = e α k t D k R t. (13)

23 The second ingredient is the smoothing power of the transpose operator R t. We denote by C b () the topological dual of C b () and by, the duality between C b () and C b (). Moreover, we identify each ν P() with an element F ν of C b () writing F µ (ϕ) := ϕ(x)µ(dx), ϕ C b (). Finally we denote by R t the transpose of R t.

24 Proposition 4 Let t > 0 and ν P(). Then R t ν << µ. Proof. Let I B(). Then we have (R tν)(i) = (R tν)(dx) = = I (R t 1l I )(x)ν(dx) = 1l I (x)(r tν)(dx) N e ta x,q t (I)ν(dx). Assume now that µ(i) = 0. Then N e ta x,q t (I) = 0 because << µ and so (R t ν)(i) = 0. N e ta x,q t In the following we shall denote by ρ ν t respect to µ. the density of R t ν with

25 Proof of Theorem 3 (i) (ii). Let u BV (, µ) and let t > 0. Then by the definition of BV function we have u(x)dk ϕ(x)µ(dx) = ϕ(x)(d k u)(dx), (14) h N, ϕ F C 1 b (). Let us first prove that R t u BV (, µ) and DR t u = e ta R tdu.

26 We have in fact by the symmetry of R t and R t Dk = e α k t Dk R t (R t u)(x)(dk ϕ)(x)µ(dx) = u(x)(r t Dk ϕ)(x)µ(dx) e α k t u(x)(dk R tϕ)(x)µ(dx) = e α k t (R t ϕ)(x)(d k u)(dx). This proves that D k (R t u) = e α k t R td k u, k N. (15)

27 We have proved that R t u BV (, µ) and D(R t u) = e ta R tdu. (16) Now we want to show that the vector measure D(R t u) can be identified with a function from L 1 (, µ). By Proposition 3 and the identity D k (R t u) = e α k t R t D ku, it follows that [D k (R t u)](dx) = e α k t ρ t (D k u)(x)µ(dx), k N. (17)

28 Now we have e ta DR t u(x) µ(dx) = e ta DR t u () and it follows easily that e ta DR t u(x) µ(dx) Du ().

29 (ii) (i). Assume that for all t > 0 we have R t u W 1,1 (, µ), e ta DR t u L 1 (, µ) and (8) holds. We recall that by Proposition 2 to show that u BV (, µ) it is enough to prove R(u) := sup m { m k=1 ud k ϕ kdµ : ϕ k C 1 b (), m i=1 ϕ 2 k (x) 1 } <. Let m N, ϕ 1,..., ϕ m Cb 1 () and m N such that m k=1 ϕ 2 k (x) 1, x.

30 Then we have m k=1 (R t u)(x) D k ϕ k(x) µ(dx) m = ϕ k (x) (D k R t u)(x)µ(dx) k=1 P m DR t u(x) µ(dx) K

31 Letting t 0 and taking supremum in ϕ 1,..., ϕ m Cb 1 () and then on m N, yields R(u) e ta R t u dµ. So, (i) follows from Proposition 2.

32 4 Non Gaussian case Consider the stochastic differential equation in a separable ilbert space dx = (AX DU(X))dt + dw (t), X(0) = x, (18) where A : D(A) is self-adjoint strictly negative (A ωi), U C 2 () is convex, DU C b (; ) and W is a cylindrical Wiener process in.

33 It is well known that equation (18) has a unique solution X(t, x). We shall denote by P t the transition semigroup, P t ϕ(x) = E[ϕ(X(t, x))], ϕ B b (). and by π t (x, ) the law of X(t, x).

34 P t has a unique invariant measure γ given by γ(dx) = ρ(x)µ(dx) where µ is the Gaussian measure of before, µ = N Q, Q = 1 2 A 1, and Z is a normalization constant. ρ(x) = Z 1 e 2U(x), x. Moreover, X(t, x) is a reversible process so that P t is symmetric.

35 Integration by parts formula The following identity can be proved easily u Dϕ, z dγ = Du, z ϕ dγ u ϕ D log ρ, z dγ + Q 1/2 z, Q 1/2 x u ϕ dγ, (19) for any u, ϕ C 1 b () and any z Q1/2 ().

36 By (19) it is not difficult to show that the gradient operator D : C 1 b () L1 (, γ; ), ϕ Dϕ, is closable in L 1 (, ν). We shall denote by W 1,1 (, γ) the domain of the closure of D and by δ(d ) the domain of the adjoint D of D in L (, γ; ).

37 Definition A function u L 1 (, γ) is said to be of bounded variation if there exists a vector measure Du M (; ) such that u(x) D F(x) γ(dx) = F(x), (Du)(dx), (20) for all F δ(d ). We denote by BV (, ν) the set of all bounded variation functions on.

38 Theorem 5 Let u BV (, γ). Then for all t > 0 we have P t u W 1,1 (, γ) and lim inf DP t u dγ Du (). (21) t 0 The proof of Theorem 5 is similar to that of Theorem 3. At the moment we have not yet proved the converse

39 As in Theorem 3, two main ingredients are needed. The first one is that P t is regular, that is all laws {π t (x, ), : x, t > 0} are mutually equivalent. In fact one can check that P t is irreducible and strong Feller. This implies that P t is regular by a theorem due to Kas minski. As a consequence, the law π t (t, dx) of X(t, x) is absolutely continuous with respect to γ.

40 Now the following result can be proved as before. Lemma Let t > 0 and let ζ M (, ). Then P t ζ << γ.

41 The second ingredient is the following commutation formula for the gradient, DP t ϕ = P t Dϕ, ϕ W 1,1 (, γ), (22) where for any t > 0, P t is defined as a bounded operator from L 1 (, ν; ) in itself P t F(x) = E[X x (t, x) F(X(t, x)], F L 1 (, ν; ). (23)

42 One can show that P t is a symmetric C 0 -semigroup on L 2 (, γ; ). Moreover, from (22) it follows by duality that D Pt F(x) = P t D F(x), x. (24)

43 Sketch of the proof of Theorem 5 We proceed as before. We first prove that P t u BV (, γ). In fact, taking into account (24) and the symmetry of P t it follows that (P t u)(x) D F(x)γ(dx) = u(x) P t D F(x)γ(dx) = u(x)d [ P t F](x)γ(dx) = This shows that P t u BV (, γ) and P t F(x), Du(dx). (25) DP t u = ( P t ) Du (26) The remaining of the proof is the same as before.