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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; 8 F 2 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 2 L1(Rn). Then (i) , (ii). (i)u 2 BV (Rn). (ii) We have Z lim jDTt u(x)jdx < 1; (1) t!0 H whereT t is the heat semigroup Z 2 1 − jx−yj Tt u(x) := p 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 2 N: We set xk = hx; ek i; k 2 N. 1 −1 We denote by A the self–ajoint operator − 2 Q . Then Aek = −αk ek ; k 2 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 2 N the following identity is well known, Z Z ∗ 1 Dk u(x) '(x) µ(dx) = u(x)Dk '(x) µ(dx); u;' 2 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∗' = −D ' + xk '; (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);' 7! D' 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 '(x) = '(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 ' 2 Bb(H) one has 1 Rt ' 2 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 ( 1 ) X jζj(K ) := sup jζ(Fk )jH :(Fk ) 2 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 jζj 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 2 M (H; H). For any h 2 N we set ζh(I) = hζ(I); ehi; 8 I 2 B(H): Then ζh is a finite measure in (H; B(H)) and we have 1 X ζ(A) = ζh(A)eh; A 2 B(H): h=1 Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Definition Let u 2 L1(H; µ). We say that u is of bounded variation (u 2 BV (H; µ)) if there exists Du 2 M (H; H) such that Z Z ∗ u(x) Dh'(x) µ(dx) = '(x)d (Dhu)(dx); H H (5) 1 8 h 2 N; 8 ' 2 F Cb (H): where (Dhu)(I) = h(Du)(I); ehi; 8 I 2 B(H): Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces A criterion to check bounded variation For any u 2 L1(H; µ) let us consider ( m Z m ) X ∗ 1 X 2 R(u) := sup uDk 'k dµ : 'k 2 Cb (H); 'k (x) ≤ 1 m k=1 H i=1 (6) It is clear that if u 2 BV (H; µ) we have R(u) ≤ jDuj(H) Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces A converse result holds Proposition 2 Letu 2 L1(H; µ). If R(u) < 1 thenu 2 BV (H; µ) and jDuj(H) ≤ R(u). Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Sketch of Proof Assume that R(u) < 1 and m 2 N. Then by (6) there exists a Rn-valued measure (D1u; :::; Dmu) such that j(D1u; :::; Dmu)j ≤ R(u): Now, a simple argument shows that u 2 BV (H; µ) and 1 X Du(I) = (Dk u)(I)ek ; 8 I 2 B(H): k=1 Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces 3 The main result Theorem 3 Letu 2 L1(H; µ). Then (i) , (ii). (i)u 2 BV (H; µ). (ii) For allt > 0 we have 1;1 −tA 1 Rt u 2 W (H; µ); je DRt ujH 2 L (H; µ) (7) and Z −tA lim inf je DRt ujH dµ < 1: (8) t!0 H Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Remark One can also show that if u 2 BV (H; µ) we have tA 0 DRt u = e Rt Du; (9) Z −tA je DRt ujdµ ≤ jDuj(H) (10) H and Z −tA lim je DRt ujdµ = jDuj(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 '(x) = e Dk '(e x + y)µt (dy); x 2 H; H we have −αk t 1 Dk Rt ' = e Rt Dk '; 8' 2 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 .
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