Fonctions on Bounded Variations in Hilbert Spaces

Fonctions on bounded variations in Hilbert spaces

Giuseppe Da Prato (SNS, Pisa)

Newton Institute, March 31, 2010

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Introduction

We recall that a function u : Rn → R is said to be of bounded variation (BV) if there exists an n-dimensional vector measure Du with ﬁnite total variation such that

Z Z 1 u(x)divF(x)dx = − hF(x), Du(dx)i, ∀ F ∈ C0 (H; H). H H

The set of all BV functions is denoted by BV (Rn). Moreover, the following result holds, see

E. De Giorgi, Ann. Mar. Pura Appl. 1954.

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Theorem 1 Letu ∈ L1(Rn). Then (i) ⇔ (ii). (i)u ∈ BV (Rn). (ii) We have Z lim |DTt u(x)|dx < ∞, (1) t→0 H whereT t is the heat semigroup

Z 2 1 − |x−y| Tt u(x) := √ e 4t u(y)dy. n 4πt R

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces As well known functions from BV (Rn) 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 studyingBV functions in general Banach or Hilbert spaces in order to extend the concepts above in an inﬁnite dimensional situation.

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces 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. Hino, JFA 2001.

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces A more analytic different approach was presented in

L. Ambrosio, M. Miranda, S. Maniglia and D. Pallara, Phisica D (to appear) and JFA, 2010.

The deﬁnition of BV functions in both approches is based on the Malliavin Sobolev space D1,1(H, µ) where H is the Cameron–Martin space and where the corresponding Mehler semigroup replaces the heat semigroup of De Giorgi’s theorem.

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces In this talk I shall present a different deﬁnition of BV function, following the paper

Ambrosio, DP, Pallara preprint 2009.

We consider a nondegenerate Gaussian measure µ in a seperable Hilbert space rather than in a Wiener space.

Moreover, our deﬁnition of BV function will involve the Sobolev space W 1,1(H, µ) instead of D1,1(H, µ) as in the previous papers.

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces 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.

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Plan of the talk

1 Basic notations and prerequisites including deﬁnition of 1,1 the Sobolev space W (H, µ) and the O. U. semigroup Rt .

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 2009.

In 4 we shall present some result of a work in progress with B. Goldys.

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces 1 Basic notations and prerequisites

H separable Hilbert space. µ non degenerate Gaussian measure of mean0 and covariance Q.

(ek ) complete orthonormal system in H and (λk ) sequence of positive numbers such that

Qek = λk ek , k ∈ N.

We set xk = hx, ek i, k ∈ N. 1 −1 We denote by A the self–ajoint operator − 2 Q . Then

Aek = −αk ek , k ∈ N, where α = 1 . k 2λk

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Integration by parts formula

For all k ∈ N the following identity is well known, Z Z ∗ 1 Dk u(x) ϕ(x) µ(dx) = u(x)Dk ϕ(x) µ(dx), u, ϕ ∈ F Cb (H), H H (2) 1 1 where F Cb (H) is set of all C functions depending only on a ﬁnite number of xk which are bounded with their derivatives and ∗ Dk is given by D∗ϕ = −D ϕ + xk ϕ, (3) k k λk

2 is the adjoint of Dk in L (H, µ).

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces By (2) it follows that the gradient operator

1 1 1 D : F Cb (H) ⊂ L (H, µ) → L (H, µ; H), ϕ 7→ Dϕ is closable, see

B. Goldys, F. Gozzi and J. Van Neerven, Pot. An. 2003.

We shall denote by W 1,1(H, µ) the domain of the closure of D in L1(H, µ).

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces The Ornstein–Uhlenbeck semigroup

We denote by Rt the Ornstein–Uhlenbeck semigroup Z tA Rt ϕ(x) = ϕ(e x + y)µt (dy), (4) H

where µt is the Gaussian measure of mean0 and covariance 2tA Qt := Q(1 − e ).

It is well known that Rt is symmetric and that µ is its unique invariant measure.

Moreover for any t > 0 and any ϕ ∈ Bb(H) one has ∞ Rt ϕ ∈ Cb (H).

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces 2 Functions of bounded variation

Let us ﬁrst recall the deﬁnition of vector measure.

A H-valued measure ζ in (H, B(H)) is a countably additive mapping ζ : B(H) → H, A 7→ ζ(A).

The total variation of ζ is the real countably additive measure on (H, B(H)) deﬁned by

( ∞ ) X |ζ|(K ) := sup |ζ(Fk )|H :(Fk ) ∈ 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.

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces By M (H, H) we mean the set of all H-valued measures with bounded total variation.

Let ζ ∈ M (H, H). For any h ∈ N we set

ζh(I) = hζ(I), ehi, ∀ I ∈ B(H).

Then ζh is a ﬁnite measure in (H, B(H)) and we have

∞ X ζ(A) = ζh(A)eh, A ∈ B(H). h=1

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Deﬁnition Let u ∈ L1(H, µ). We say that u is of bounded variation (u ∈ BV (H, µ)) if there exists Du ∈ M (H, H) such that Z Z ∗ u(x) Dhϕ(x) µ(dx) = ϕ(x)d (Dhu)(dx), H H (5) 1 ∀ h ∈ N, ∀ ϕ ∈ F Cb (H). where (Dhu)(I) = h(Du)(I), ehi, ∀ I ∈ B(H).

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces A criterion to check bounded variation

For any u ∈ L1(H, µ) let us consider

( m Z m ) X ∗ 1 X 2 R(u) := sup uDk ϕk dµ : ϕk ∈ Cb (H), ϕk (x) ≤ 1 m k=1 H i=1 (6)

It is clear that if u ∈ BV (H, µ) we have

R(u) ≤ |Du|(H)

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces A converse result holds Proposition 2 Letu ∈ L1(H, µ). If R(u) < ∞ thenu ∈ BV (H, µ) and |Du|(H) ≤ R(u).

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Sketch of Proof

Assume that R(u) < ∞ and m ∈ N. Then by (6) there exists a Rn-valued measure (D1u, ..., Dmu) such that

|(D1u, ..., Dmu)| ≤ R(u).

Now, a simple argument shows that u ∈ BV (H, µ) and

∞ X Du(I) = (Dk u)(I)ek , ∀ I ∈ B(H). k=1

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces 3 The main result

Theorem 3 Letu ∈ L1(H, µ). Then (i) ⇔ (ii). (i)u ∈ BV (H, µ). (ii) For allt > 0 we have

1,1 −tA 1 Rt u ∈ W (H, µ), |e DRt u|H ∈ L (H, µ) (7)

and Z −tA lim inf |e DRt u|H dµ < ∞. (8) t→0 H

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Remark One can also show that if u ∈ BV (H, µ) we have

tA 0 DRt u = e Rt Du, (9)

Z −tA |e DRt u|dµ ≤ |Du|(H) (10) H and Z −tA lim |e DRt u|dµ = |Du|(H). (11) t→0 H

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Basic ingredients of the proof of Theorem 3

The ﬁrst ingredient is an elementary formula for the commutator between Rt and Dk . Since Z −αk t tA Dk Rt ϕ(x) = e Dk ϕ(e x + y)µt (dy), x ∈ H, H

we have

−αk t 1 Dk Rt ϕ = e Rt Dk ϕ, ∀ϕ ∈ Cb (H). (12)

Since Rt is symmetric we deduce by duality the identity

∗ −αk t ∗ Rt Dk = e Dk Rt . (13)

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces The second ingredient is the smoothing power of the transpose 0 operator Rt .

0 We denote by Cb(H) the topological dual of Cb(H) and by h·, ·i 0 the duality between Cb(H) and Cb(H) .

Moreover, we identify each ν ∈ P(H) with an element Fν of 0 Cb(H) writing Z Fµ(ϕ) := ϕ(x)µ(dx), ∀ ϕ ∈ Cb(H). H

0 Finally we denote by Rt the transpose of Rt .

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Proposition 4 0 Lett > 0 and ν ∈ P(H). ThenR t ν << µ.

Proof. Let I ∈ B(H). Then we have Z Z 0 0 0 (Rt ν)(I) = (Rt ν)(dx) = 1lI(x)(Rt ν)(dx) I H Z Z = (R 1l )(x)ν(dx) = N tA (I)ν(dx). t I e x,Qt H H

Assume now that µ(I) = 0. Then N tA (I) = 0 because e x,Qt 0 N tA << µ and so (R ν)(I) = 0. e x,Qt t

ν 0 In the following we shall denote by ρt the density of Rt ν with respect to µ.

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Proof of Theorem 3

(i) ⇒ (ii). Let u ∈ BV (H, µ) and let t > 0. Then by the deﬁnition of BV function we have Z Z ∗ u(x)Dk ϕ(x)µ(dx) = ϕ(x)(Dk u)(dx), (14) H H

1 ∀ h ∈ N, ∀ ϕ ∈ F Cb (H).

Let us ﬁrst prove that Rt u ∈ BV (H, µ) and

tA 0 DRt u = e Rt Du.

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces ∗ −αk t ∗ We have in fact by the symmetry of Rt and Rt Dk = e Dk Rt

Z Z ∗ ∗ (Rt u)(x)(Dk ϕ)(x)µ(dx) = u(x)(Rt Dk ϕ)(x)µ(dx) H H Z Z −αk t ∗ −αk t e u(x)(Dk Rt ϕ)(x)µ(dx) = e (Rt ϕ)(x)(Dk u)(dx). H H

This proves that

−αk t 0 Dk (Rt u) = e Rt Dk u, ∀ k ∈ N. (15)

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces We have proved that Rt u ∈ BV (H, µ) and

tA 0 D(Rt u) = e Rt Du. (16)

Now we want to show that the vector measure D(Rt u) can be identiﬁed with a function from L1(H, µ).

−αk t 0 By Proposition 3 and the identity Dk (Rt u) = e Rt Dk u, it follows that

−αk t [Dk (Rt u)](dx) = e ρt (Dk u)(x)µ(dx), ∀ k ∈ N. (17)

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces 1,1 (ii) ⇒ (i). Assume that for all t > 0 we have Rt u ∈ W (H, µ), −tA 1 e DRt u ∈ L (H, µ) and (8) holds.

We recall that by Proposition 2 to show that u ∈ BV (H, µ) it is enough to prove

( m Z m X ∗ 1 X 2 o R(u) := sup uDk ϕk dµ : ϕk ∈ Cb (H), ϕk (x) ≤ 1 < ∞. m k=1 H i=1

1 Let m ∈ N, ϕ1, ..., ϕm ∈ Cb (H) and m ∈ N such that

m X 2 ϕk (x) ≤ 1, ∀ x ∈ H. k=1

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Then we have m Z X ∗ (Rt u)(x) Dk ϕk (x) µ(dx) k=1 H

m X Z = ϕk (x)(Dk Rt u)(x)µ(dx) k=1 H Z ≤ |PmDRt u(x)|µ(dx) K

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces 1 Letting t → 0 and taking supremum in ϕ1, ..., ϕm ∈ Cb (H) and then on m ∈ N , yields Z −tA R(u) ≤ |e Rt u|dµ. H

So, (i) follows from Proposition 2.

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces 4 Non Gaussian case

Consider the stochastic differential equation in a separable Hilbert space H   dX = (AX − DU(X))dt + dW (t), (18)  X(0) = x,

where A : D(A) ⊂ H → H is self-adjoint strictly negative 2 (A ≤ −ωI), U ∈ C (H) is convex, DU ∈ Cb(H; H) and W is a cylindrical Wiener process in H.

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces It is well known that equation (18) has a unique solution X(t, x).

We shall denote by Pt the transition semigroup,

Pt ϕ(x) = E[ϕ(X(t, x))], ϕ ∈ Bb(H).

and by πt (x, ·) the law of X(t, x).

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Pt has a unique invariant measure γ given by

γ(dx) = ρ(x)µ(dx)

where µ is the Gaussian measure of before, µ = NQ, 1 −1 Q = − 2 A , and

ρ(x) = Z −1e−2U(x), x ∈ H. Z is a normalization constant.

Moreover, X(t, x) is a reversible process so that Pt is symmetric.

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Integration by parts formula

The following identity can be proved easily Z Z u hDϕ, zi dγ = − hDu, zi ϕ dγ H H (19) Z Z − u ϕ hD log ρ, zi dγ + hQ−1/2z, Q−1/2xi u ϕ dγ, H H

1 1/2 for any u, ϕ ∈ Cb (H) and any z ∈ Q (H).

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces By (19) it is not difﬁcult to show that the gradient operator

1 1 D : Cb (H) → L (H, γ; H), ϕ 7→ Dϕ, is closable in L1(H, ν).

We shall denote by W 1,1(H, γ) the domain of the closure of D and by δ(D∗) the domain of the adjoint D∗ of D in L∞(H, γ; H).

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Deﬁnition A function u ∈ L1(H, γ) is said to be of bounded variation if there exists a vector measure Du ∈ M (H; H) such that Z Z u(x) D∗F(x) γ(dx) = hF(x), (Du)(dx)i, (20) H H for all F ∈ δ(D∗).

We denote by BV (H, ν) the set of all bounded variation functions on H.

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Theorem 5 1,1 Letu ∈ BV (H, γ). Then for allt > 0 we haveP t u ∈ W (H, γ) and Z lim inf |DPt u|dγ ≤ |Du|(H). (21) t→0 H

The proof of Theorem 5 is similar to that of Theorem 3. At the moment we have not yet proved the converse

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces As in Theorem 3, two main ingredients are needed.

The ﬁrst one is that Pt is regular, that is all laws

{πt (x, ·), : x ∈ H, t > 0} are mutually equivalent.

In fact one can check that Pt is irreducible and strong Feller.

This implies that Pt is regular by a theorem due to Kas’minski.

As a consequence, the law πt (t, dx) of X(t, x) is absolutely continuous with respect to γ.

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Now the following result can be proved as before. Lemma 0 Lett > 0 and let ζ ∈ M (H, H). ThenP t ζ << γ.

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces The second ingredient is the following commutation formula for the gradient,

1,1 DPt ϕ = Pbt Dϕ, ϕ ∈ W (H, γ), (22) where for any t > 0, Pbt is deﬁned as a bounded operator from L1(H, ν; H) in itself

∗ 1 Pbt F(x) = E[Xx (t, x) F(X(t, x)], F ∈ L (H, ν; H). (23)

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces One can show that Pbt is a symmetric C0-semigroup on L2(H, γ; H).

Moreover, from (22) it follows by duality that

∗ ∗ D Pbt F(x) = Pt D F(x), x ∈ H. (24)

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Sketch of the proof of Theorem 5

We proceed as before. We ﬁrst prove that Pt u ∈ BV (H, γ). In fact, taking into account (24) and the symmetry of Pt it follows that Z Z ∗ ∗ (Pt u)(x) D F(x)γ(dx) = u(x) Pt D F(x)γ(dx) H H (25) Z Z ∗ = u(x)D [Pbt F](x)γ(dx) = hPbt F(x), Du(dx)i. H H

This shows that Pt u ∈ BV (H, γ) and

0 DPt u = (Pbt ) Du (26)

The remaining of the proof is the same as before.

Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces