BV functions in Hilbert spaces.

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Introduction

Let H be a separabile Hilbert space and ν a Borel probability measure on H. Our main assumption is the following Hypothesis 1 (DP–Debussche). There exists R ∈ L(H) symmetric and positive such that for all p > 1 there is Cp > 0 such that Z 1 hRDϕ(x), zi ν(dx) ≤ Cp |z| kϕkLp(H,ν), ∀ ϕ ∈ Cb (H), ∀ z ∈ H. H

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Hypothesis 1 is clearly fulfilled when ν = NQ is Gaussian with R = Q1/2. Moreover, has been proved by DP-Debussche when: • ν is the invariant measure of the Burgers equation, Ann. Poincaré 2016 • ν is the invariant measure for a family of Reaction–Diffusion equations. J. Ev. Equ, 2017 4 • ν is the invariant measure of the Φ2 model. Paper in preparation

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Closability of M

Under Hypothesis 1 it is not difficult to show that M := RD is closable in Lp(H, ν) for all p ∈ (1, ∞). 1,p We denote by Mp the closure of M, by W (H, ν) its domain, ∗ by Mp its adjoint. By definition Z Z ∗ 1,p ∗ hMp u, Fi dν = u Mp (F) dν, ∀ u ∈ W (H, ν), F ∈ D(Mp ). H H When no confusion can arise, we shall omit the sub–index p. M is the natural generalisation of the Malliavin derivative and M∗ of the Gaussian divergence, or Skorokhod integral, for the probability measure ν.

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. BV functions with respect to ν

Let ν be a Borel probability measure on H, p > 1 and u ∈ Lp(H, ν). We say that u belongs to BV (H, ν) if there exists a vector measure m : B(H) → H with finite total variation such that, Z Z u M∗(F) dν = hF, dmi, ∀ F ∈ E , H H where E is the following space of test functions

 Pn 1 E := F = i=1 ui zi , n ∈ N, ui ∈ F Cb (H), zi ∈ H

1 1 and F Cb (H) is the subset of Cb (H) of functions depending only in a finite number of variables.

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Perimeters

We say that a Borel subset A of H has a finite perimeter with respect to ν if 1A ∈ BV (H, ν).

In this case there is a vector measure mA with finite total variation such that Z Z ∗ M (F) dν = hF, dmAi, ∀ F ∈ E . A H

mA is called the perimeter measure of A and its total variation

|mA|(H)

the perimeter of A.

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Outline

(i) In the first part of the talk we shall show, following

DP–Lunardi, ArXiv 2018,

that a function u ∈ Lp(H, ν) (for some p > 1) belongs to BV (H, ν) if and only if there is K > 0 such that Z ∗ u M (F) dν ≤ K kFk∞, ∀ F ∈ E . (BV ) H

where E is the space of test functions introduced before.

 Pn 1 E := F = i=1 ui zi , n ∈ N, ui ∈ F Cb (H), zi ∈ H .

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. (ii) The second part will be devoted to examples of sets of finite perimeters. We shall concentrate on the sub–level sets of functions g : H → R. A particular attention will be payed to the special function

g : H → (−∞, 0], g = inf ξ(t), t∈[0,1]

where dξ = b(ξ)dt + dBt , ξ(0) = 0, which arises in reflexion problems. This fact is well know when b = 0 (in this case ξ reduces to the Brownian motion) thanks to the reflexion principle. The general case seems to be open.

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Some comments about literature

Functions of bounded variation in abstract Wiener spaces, have been first studied by M. Fukushima and M. Hino in JFA-2000 by using Dirichlet forms. Then L. Ambrosio, S. Maniglia, M. Miranda and D. Pallara see JFA-2010, have improved these results (always in abstract Wiener spaces) using purely analytical tools, in particular, disintegration of Gaussian measures. The tool of disintegration seems difficult to handle for non Gaussian measures. We were able instead to generalise the Riesz theorem in H, thanks to a suitable integration by parts formula which is a consequence of Hypothesis 1. For abstract results for a function ν possessing the Fomin derivative in enough many directions see V. Bogachev, AMS 2010 and his school.

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Summary on vector measures

A vector measure m in H is a σ–additive mapping

m : B(H) → H, B 7→ µ(B).

We define the total variation of m setting

∞ X |m|(I) := sup |m(Ik )|, ∀ I ∈ B(H), (Ik )⊂ΠI k=1

where ΠI denotes the set of all decompositions of I. One can show that |m| is a positive measure (not necessarily finite). If |m|(H) < ∞ we say that m has a finite total variation and we call |m|(H) the total variation of m.

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. We shall denote by M (H, H) the set of all vector measures with finite total variation. Here is a simple example. Example Let ν be a scalar Borel measure andF ∈ L1(H, ν; H). Then

m(dx) := F(x) ν(dx)

is a vector measure whose total variation is Z |m|(H) = kF(x)k ν(dx). H

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Let m ∈ M (H, H) and let (eh) be an orthonormal basis in H. Then we define the projection mh of m to be the scalar measure

mh(B) = hm(B), ehi, ∀ B ∈ B(H), h ∈ N.

It is easy to see that

P∞ m(B) = h=1 mh(B) eh, ∀ B ∈ B(H).

Conversely, given a sequence (mh) of signed measures, we can construct a vector measure

P∞ m(B) := h=1 µh(B) eh, ∀ B ∈ B(H),

provided sup |mn|(H) < ∞. n→∞

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. A basic integration by parts formula

Proposition (i) Constant vectors fields belong toD (M∗). As a consequence, p for anyz ∈ H there existsv z ∈ L (H, ν), ∀ p ∈ [1, ∞), such that ∗ M (z) = vz . (ii) Moreover, the followig integration by parts formula holds Z Z 1 hMϕ, zi dν = vz ϕ dν, ∀ ϕ ∈ Cb (H), (IBP) H H

vz coincides with the Fomin derivative of ν in the direction Rz.

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Proof

Fix z ∈ H and consider the constant vector field

F(x) = z, ∀ x ∈ H.

Then by Hypothesis 1, for all p > 1 there exists Cp > 0 such that Z 1 hMϕ(x), F(x)i ν(dx) ≤ Cp |z| kϕkLp(H,ν), ∀ ϕ ∈ Cb (H). H

∗ p ∗ This implies F ∈ D(Mq ) , q = p−1 , and setting Mq (F) = vz we q have vz ∈ L (H, ν) and (IBP) follows by the arbitrariness of p.

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Remark Replacing ϕ by u ϕ we find the following useful generalisation of (IBP), for all z ∈ H, Z Z uhMϕ, zi dν = − ϕhMu, zi dµ H H Z 1 + vz u ϕ dν, ∀ u, ϕ ∈ Cb (H). H (IBP1)

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. The basic characterisation result

Theorem (DP-Lunardi-ArXiv-2018) Assume Hypothesis 1 and letu ∈ Lp(H, ν) for somep > 1. Thenu ∈ BV (H, ν) if and only if there isK > 0 such that Z ∗ u M (F) dν ≤ K kFk∞, ∀ F ∈ E . (BV ) H

where E is the space of test functions defined before

 Pn 1 E := F = i=1 ui zi , n ∈ N, ui ∈ F Cb (H), zi ∈ H .

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Sketch of the proof. We claim that there exists m ∈ M (H, H) such that, Z Z u M∗(F) dν = hF, dmi, ∀ F ∈ E . H H

First we fix an orthonormal basis (eh) on H and for any h ∈ N we show, by a suitable generalisation of Riesz’s theorem, the existence of a scalar measure mh, such that (thanks to (IBP1)) 1 for all ϕ ∈ Cb (H) we have Z Z Z ∗ u M (ϕ eh) dν = u (−hMϕ, ehi + ϕ veh ) dν = ϕ dmh. H H H Then we shall define m ∈ BV (H, ν) as

Pn m(B) := h=1 µh(B) eh, ∀ n ∈ N, B ∈ B(H),

after proving that sup |mn|(H) < ∞. n→∞

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Sets of finite perimeter

A Borel set A ⊂ H has a finite perimeter if 1A ∈ BV (H, ν). In this case there exists a vector measure mA ∈ M (H, H) such that Z Z ∗ M (F) dν = hF, dmAi, ∀ F ∈ E . A H

By Theorem 1, 1A belongs to BV (H, ν) if and only if there exists K > 0 such that Z ∗ M (F) dν ≤ K kFk∞, ∀ F ∈ E . A

When A = {g ≥ r}, where g : H → R, the following lemma provides an useful expression for Z M∗(F) dν. {g≥r}

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Lemma , Assume thatg ∈ W 1 2(H, ν) and letz ∈ H. Then for allr ∈ R we have Z 1 Z M∗(F) dν = lim hMg, Fi dν, ∀ F ∈ E . {g≥r} →0 2 {r−≤g≤r+}

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Proof

Fix  > 0 and apply the identity Z Z hMϕ, Fi dν = ϕ M∗(F) dν H H

where F ∈ E and ϕ = θ(g), where

 0 if ξ ≤ r −   ξ−r θ(ξ) =  if ξ ∈ [r − , r + ]  1 if ξ ≥ r + 

1,2 Then it follows that θ(g) ∈ W (H, ν) and

0 M(θ(g)) = θ(g)Mg.

see DP-Lunardi-Tubaro, Trans. AMS 2018,

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Since  0 if ξ ≤ r −  0  1 θ(ξ) =  if ξ ∈ (r − , r + ]  0 if ξ > r + , we have 1 Z Z hMg, Fi ϕ dν = M∗(F) dν, 2 {r≤g≤r−} {g≥r−}

and as  → 0 1 Z Z lim hMg, Fi ϕ dν = M∗(F) dν. →0 2 {r−≤g≤r−} {g≥r}

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Proposition Assume thatg ∈ W 1,2(H, ν) and Mg ∈ L∞(H, ν). Then {g ≥ r} has a finite perimeter for almost allr ∈ I.

Proof. From the lemma we have Z ∗ 1 −1 M (F) dν ≤ kFk∞ kMgk∞ lim inf (ν◦g )([r −, r +]). {g≥r} →0 2 Set H(r) = ν({g ≥ r}) = (ν ◦ g−1)((−∞, r]). Note that H is increasing and so, right differentiable for almost all r ∈ [−∞, 0). For such an r we have Z ∗ + M (F) dν ≤ kFk∞ D H(r) {g≥r} so that {g ≥ r} has a finite perimeter for almost all r ∈ I.

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. But it is important to prove that the perimeter is finite for all r.

A typical case is when a surface measure σr can be defined with respect to ν. This happens if the following holds Hypothesis 2 Letg ∈ W 1,2(H, ν) and

Mg ∈ D(M∗). kMgk2

A similar hypothesis was introduced in Airault–Malliavin, Bull. Sci. Math. 88, when ν is Gaussian. Let us recall the following result, proved in DP-Lunardi-Tubaro, Trans. AMS 2018.

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Theorem Assume that ν fulfills Hypotheses 1 and 2. Then for all 1,2 ϕ ∈ Cb(H) ∪ W (H, µ) and for allr ∈ R there exists the limit Z Z ν 1 ϕ dσr := lim ϕ dν. {g=r} →0  {r−≤g≤r+}

As a consequence {g ≥ r} has a finite perimeter for allr < 0.

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Example Let ν be the invariant measure of the Burgers, a reaction 4 diffusion equation or of the Φ2 model. Then balls and half–spaces (and several other sets) have finite perimeters.

See DP-Lunardi-Tubaro, Trans. AMS, 2018.

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Hypothesis 2 is too strong for several applications. The next proposition provides a sufficient condition implying that the perimeter of {g ≥ r} is finite for all r ∈ (−∞, 0).

Proposition Assume thatg ∈ W 1,2(H, ν), Mg ∈ L∞(H, ν), ν ◦ g−1 is absolutely continuous with respect to the Lebesgue measure in (−∞, 0) and that the density

d(ν ◦ g−1) (r) = ρ(r), r ∈ H, dλ is continuous. Then {g ≥ r} has a finite perimeter for all r ∈ (−∞, 0).

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Proof. Recall that Z 1 Z M∗(F) dν = lim hMg, Fi dν, ∀ F ∈ E . {g≥r} →0 2 {r−≤g≤r+}

Let r ∈ (−∞, 0) and ρ ≤ C near r. Then Z Z ∗ 1 1 M (F) dν ≤ kMgk∞ kFk∞ lim inf dν {g≥r} 2 →0 2 {r−

1 Z r+ ≤ kMgk∞ kFk∞ lim inf ρ(s)ds ≤ ρ(0)kMgk∞ kFk∞. →0  r− So, we can apply the last Theorem and the conclusion follows.

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. (ii) A special g

Now we consider a one dimensional SDE

dξ = b(ξ)dt + dB(t), ξ(0) = 0,

2 where b ∈ Cb (R), moreover Z η v(η) = b(r)dr, η ∈ R, 0

is bounded and B(·) is a Brownian motion in (Ω, F , P). We denote by ν the law of ξ(·) in C([0, 1]).

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Theorem (BoDaLuTu) The law of g := min ξ(t) t∈[0,1] in (−∞, 0] is absolutely continuous with respect to the Lebesgue measure with a continuous density. Therefore {g ≥ r} has a finite perimeter for allr ∈ (−∞, 0).

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Proof

−1 Step 1. We show that P ◦ (ξ(·) ) << NQ and d ◦ (ξ(·)−1) P = Ψ, dNQ with Ψ smooth. In fact, let us consider the Girsanov transform

dQ = ρ(1)dP, where − 1 R 1 |b(ξ(s))|2ds−R 1 b(ξ(s)) dB(s) ρ(1) = e 2 0 0 . Then ξ(·) is a Brownian motion in (Ω, F , Q). −1 Therefore Q ◦ (ξ(·) ) = NQ, and so, for any F : C([0, 1]) → R bounded and Borel one has Z Z F(ξ(·)) dQ = F(h) NQ(dh). Ω C([0,1])

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. We note that by Ito’s formula applied to Z η v(η) = b(r)dr, η ∈ R, 0 we have

−v(ξ(1))+ 1 R 1 b2(ξ(s))ds+ 1 R 1 b0(ξ(s))ds ρ(1) = e 2 0 2 0 .

Since −1 EP(F(ξ(·))) = EQ(F(ξ(·))ρ(1) ), we have Z Z F(ξ(·)) dP = F(h(·))Ψ(h) NQ(dh), Ω C([0,1] where

v(h(1))− 1 R 1 b2(h(s))ds− 1 R 1 b0(h(s))ds Ψ(h) = e 2 0 2 0 .

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. −1 So, P ◦ (ξ(·)) = Ψ NQ and Step 1 is proved. In particular, we have Z Z α( min ξ(·)) dP = α( min h(·))Ψ(h) NQ(dh), Ω t∈[0,1] C([0,1] t∈[0,1]

for any α : R → R bounded and Borel. −1 Claim. Ψ NQ ◦ g << λ and

d(Ψ N ◦ g−1) Q (dr) =: ρ (r)dr dλ Ψ

with ρΨ(r) continuous on r. So,

Z Z 0 α( min ξ(·)) dP = α( min )ρΨ(r)dr, Ω t∈[0,1] −∞ t∈[0,1]

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. This shows that the law of mint∈[0,1] ξ(·) is absolutely continuous with respect to the Lebesgue measure λ on (−∞, 0) with a continuos density proving the theorem. Finally, the claim follows thanks to some recent results from Florit and Nualart, Stat. Probab Lett. 15. because, these results make possible to construct a surface integral for mint∈[0,1] h(t) with respect to NQ. See Bonaccorsi, DP and Tubaro, J. Evol. Equ. 18.

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Recalling some results on surface integrals

Let X be a separable Banach space, µ a Gaussian measure on X and g : X → R Borel. For any r ∈ I set Z Fr (ϕ) = ϕ dν, ϕ ∈ Cb(H). {g≥r}

If for all ϕ ∈ Cb(H) there exists the limit Z 0 1 Fr (ϕ) = lim ϕ dν →0 2 r−≤g≤r+

g and a probability measure σr in H such that Z 0 g Fr (ϕ) = ϕ σr , H

g we say that σr is a surface measure with respect to g and ν.

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. The theory of surface measures in an infinite dimensional space was initiated by Airault and Malliavin, see [AiMa88] for Gaussian measures and developed by several authors. However, in all these papers some regularity of g are assumed. In particular that Z |Mg|−pdν < ∞. X This assumption is not fulfilled in the case under consideration

g = min ξ(t) t∈[0,1]

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. The following weaker assumption was given in Nualart [Nu06] Hypothesis 2 Giveng : X → I belonging to the domain of the Malliavin derivative, there exist two random variablesu : X → X and γ : X → R such that

hMg(x), u(x)i = γ(x), ∀ x ∈ g−1(I)

and u ∈ D(M∗) ∀ p ≥ 1, γ p ∗ whereM p is the Malliavin derivative andM p its dual.

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Remark This assumption is fulfilled when

g = inf B(t), t∈[0,1]

where B(·) is a standard Brownian motion, see Florit and Nualart, Stat. Probab Lett. 15 and Bonaccorsi, DP and Tubaro, J. Evol. Equ. 18.

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces. Theorem Under Hypothesis 2 for everyr ∈ I there exists a unique Borel g measure σr onH Moreover, Z 0 g Fϕ(r) = ρϕ(r) = ϕ(x) σr (dx), ∀ ϕ ∈ UCb(X). X is continuous and d(ϕ µ) ◦ g−1 (r) = F 0 (r). dλ ϕ

See Bonaccorsi–DP–Tubaro, J. Evol. Equ. 2018.

Giuseppe Da Prato Scuola Normale Superiore, Pisa, Italy BV functions in Hilbert spaces.