A class of infinite dimensional stochastic processes

Transcription

1 A class of infinite dimensional stochastic processes John Karlsson Linköping University CRM Barcelona, July 7, 214 Joint work with Jörg-Uwe Löbus John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

2 1 Outline 2 The Dirichlet form 3 Closability 4 Quasi-regularity 5 Moving to a geometric setting 6 References John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

3 Coercive forms A bilinear form E : H H R is called coercive if E(x, x) C x H, x H. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

4 Coercive forms A bilinear form E : H H R is called coercive if E(x, x) C x H, x H. A coercive closed form is called a Dirichlet form if (i) u + 1 D(E), (ii) E(u + u + 1, u u + 1), (iii) E(u u + 1, u + u + 1). John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

5 Coercive forms A bilinear form E : H H R is called coercive if E(x, x) C x H, x H. A coercive closed form is called a Dirichlet form if (i) u + 1 D(E), (ii) E(u + u + 1, u u + 1), (iii) E(u u + 1, u + u + 1). For symmetric forms E(u + 1, u + 1) E(u, u). John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

6 Connection to stochastic processes In the 197s Fukushima showed that if a Dirichlet form on a locally compact state space is regular one can construct an associated Markov process. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

7 Connection to stochastic processes In the 197s Fukushima showed that if a Dirichlet form on a locally compact state space is regular one can construct an associated Markov process Röckner and Ma solved the question in an infinite dimensional setting: John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

8 Connection to stochastic processes In the 197s Fukushima showed that if a Dirichlet form on a locally compact state space is regular one can construct an associated Markov process Röckner and Ma solved the question in an infinite dimensional setting: Röckner-Ma A Dirichlet form on a separable metric space is associated with a Markov process with decent sample paths if and only if the form is quasi-regular. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

9 The Dirichlet form We are concerned with Dirichlet forms of type E(F, G) = DF, ADG H ϕdν = DF, λ i S i, DG H S i ϕdν. i=1 H John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

10 The Dirichlet form We are concerned with Dirichlet forms of type E(F, G) = DF, ADG H ϕdν = DF, λ i S i, DG H S i i=1 H ϕdν. Here H is the space of absolutely continuous R d -valued functions on [, 1], ν is the Wiener measure and ϕ is a weight function. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

11 Closability definition Definition A bilinear form is closable if it has a closed extension. More precisely, let E be a positive definite bilinear form on H with domain D(E). (E, D(E)) is called closable on H if for all u n D(E), n N such that E(u n u m, u n u m ) and m,n u n in H, we have E(u n, u n ). John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

12 Framework We study the form on the set of functions { Z := F (γ) = f (γ(s 1 ),..., γ(s k )), γ Ω : } < s 1 < < s k = 1, f Cp (R d k ), k N John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

13 Framework We study the form on the set of functions { Z := F (γ) = f (γ(s 1 ),..., γ(s k )), γ Ω : } < s 1 < < s k = 1, f Cp (R d k ), k N where f Cp means that f and all its partial derivatives are smooth with polynomial growth. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

14 Framework We study the form on the set of functions { Z := F (γ) = f (γ(s 1 ),..., γ(s k )), γ Ω : } < s 1 < < s k = 1, f Cp (R d k ), k N where f Cp means that f and all its partial derivatives are smooth with polynomial growth. As well as { Y := F (γ) = f (γ(s 1 ),..., γ(s k )), γ Ω : F Z, s 1,..., s k { l 2 : l {1,..., 2 n } } }, n N. n John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

15 Closability result Lemma If f ϕ L1 (ν), f Z. (1) Then the operators D, and δ are closable as operators L 2 (ϕν) Z L 2 (ϕν; H) and L 2 (ϕν; H) Z H L 2 (ϕν) respectively. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

16 Equivalence classes x := {x + c1 : c R}, L 2 (ϕν) 1 := {(x) : x L 2 (ϕν), δ(z) := The equivalence class of δ(z), z Z H. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

17 Equivalence classes x := {x + c1 : c R}, L 2 (ϕν) 1 := {(x) : x L 2 (ϕν), δ(z) := The equivalence class of δ(z), z Z H. Lemma If f ϕ L1 (ν), f Z. (1) Then the operators D 1, and δ 1 are closable as operators L 2 (ϕν; H) Z 1 L 2 (ϕν) 1 and L 2 (ϕν) 1 Z 1 H L2 (ϕν) respectively. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

18 Closability of the form Theorem Let ϕ satisfy (1) and assume that A ε Id for some ε >. (a) The form (E, Y ) is closable in L 2 (ϕν). Let (E, D Y (E)) denote the closure of (E, Y ) in L 2 (ϕν). (b) If sup c 1,c 2,... {,1} d j=1 p= λip,j ϕdν 2 p <, (2) then (E, Z) is closable and we let (E, D Z (E)) denote the closure of (E, Z) on L 2 (ϕν). Furthermore D Z (E) = D Y (E) under (2). (c) Z D Y (E) if and only if (2). (d) If (E, Z) is closable in L 2 (ϕν) then (2) holds. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

19 In the special case Remark If the sequence of eigenvalues is non-decreasing < λ 1 (γ) < λ 2 (γ),..., then relation (2) simplifies to λd2 p ϕdν 2 p <. p= John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

20 E-nests Definition (i) An increasing sequence (F k ) k N of closed subsets of E is called an E-nest if k 1 D(E) Fk is dense in D(E) w.r.t. Ẽ1, where D(E) Fk denotes {u D(E) : u = m-a.e. on E \ F k }. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

21 E-nests Definition (i) An increasing sequence (F k ) k N of closed subsets of E is called an E-nest if k 1 D(E) Fk is dense in D(E) w.r.t. Ẽ1, where D(E) Fk denotes {u D(E) : u = m-a.e. on E \ F k }. (ii) A subset N E is called E-exceptional if N k Fk c for some E-nest (F k ) k N. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

22 E-nests Definition (i) An increasing sequence (F k ) k N of closed subsets of E is called an E-nest if k 1 D(E) Fk is dense in D(E) w.r.t. Ẽ1, where D(E) Fk denotes {u D(E) : u = m-a.e. on E \ F k }. (ii) A subset N E is called E-exceptional if N k Fk c for some E-nest (F k ) k N. (iii) We say that a property holds E-quasi-everywhere if it holds everywhere outside some E-exceptional set. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

23 E-nests Definition (i) An increasing sequence (F k ) k N of closed subsets of E is called an E-nest if k 1 D(E) Fk is dense in D(E) w.r.t. Ẽ1, where D(E) Fk denotes {u D(E) : u = m-a.e. on E \ F k }. (ii) A subset N E is called E-exceptional if N k Fk c for some E-nest (F k ) k N. (iii) We say that a property holds E-quasi-everywhere if it holds everywhere outside some E-exceptional set. Definition An E-quasi-everywhere defined function f is called E-quasi-continuous if there exists an E-nest (F k ) k N such that f is continuous on (F k ) k N. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

24 Definition of quasi-regularity Definition A Dirichlet form (E, D(E)) on L 2 (E, m) is called quasi-regular if: (i) There exists an E-nest (F k ) k N consisting of compact sets, (ii) There exists an Ẽ 1/2 1 -dense subset of D(E) whose elements have E-quasi-continuous m-versions, (iii) There exist u n D(E), n N, having E-quasi-continuous m-versions ũ n, n N, and an E-exeptional set N E such that {ũ n : n N} separates the points of E \ N. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

25 The weight function ϕ We consider weight functions ϕ + and ϕ of the form { 1 ϕ + (γ) := exp b s (γ), dγ s R d } b s (γ) 2 ds, (3) John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

26 The weight function ϕ We consider weight functions ϕ + and ϕ of the form { 1 ϕ + (γ) := exp b s (γ), dγ s R d } b s (γ) 2 ds, (3) and { 1 ϕ (γ) := exp b s (γ), dγ s R d } b s (γ) 2 ds, (4) where b s (γ) is adapted to the natural filtration of the Wiener process. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

27 The weight function ϕ We consider weight functions ϕ + and ϕ of the form { 1 ϕ + (γ) := exp b s (γ), dγ s R d } b s (γ) 2 ds, (3) and { 1 ϕ (γ) := exp b s (γ), dγ s R d } b s (γ) 2 ds, (4) where b s (γ) is adapted to the natural filtration of the Wiener process. Under the condition f ϕ + L1 (ν), and f ϕ L1 (ν), (5) John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

28 The weight function ϕ We consider weight functions ϕ + and ϕ of the form { 1 ϕ + (γ) := exp b s (γ), dγ s R d } b s (γ) 2 ds, (3) and { 1 ϕ (γ) := exp b s (γ), dγ s R d } b s (γ) 2 ds, (4) where b s (γ) is adapted to the natural filtration of the Wiener process. Under the condition f ϕ + L1 (ν), and f ϕ L1 (ν), (5) for all f Z, the earlier closability results hold for ϕ + as well as for ϕ. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

29 The Novikov condition Lemma Let ϕ +, and ϕ be defined as in (3), and (4). If (5) holds then we have the Novikov condition [ { 1 1 }] E exp b s (γ) 2 ds <. 2 In particular, E[ϕ + ] = E[ϕ ] = 1. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

30 The Novikov condition Lemma Let ϕ +, and ϕ be defined as in (3), and (4). If (5) holds then we have the Novikov condition [ { 1 1 }] E exp b s (γ) 2 ds <. 2 In particular, E[ϕ + ] = E[ϕ ] = 1. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

31 Main theorem Theorem For ϕ = ϕ + and for ϕ = ϕ, the closure of E(F, F ) = λ i S i, DF 2 H ϕdν, F Z, i=1 in L 2 (ϕν), is quasi-regular. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

32 Geometric framework Let M be a connected geometrically, stochastically complete, and torsion free manifold. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

33 Geometric framework Let M be a connected geometrically, stochastically complete, and torsion free manifold. We study the form Ê( ˆF, Ĝ) = ˆD ˆF, A ˆDĜ ˆϕd ˆν H = ˆD ˆF, λ i S i, ˆDĜ i=1 H S i H ˆϕd ˆν, where ˆν is the Wiener measure on the path space P m (M) := {ˆγ C([, 1]; M) : ˆγ() = m }, m M and ˆϕ is a weight function to be specified. (6) John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

34 Geometric framework cont. More precisely let P (R d ) Ω := {γ C([, 1]; R d ) : γ() = }, I : P (R d ) P m (M) be the Itô map and ˆν be the image measure of the Wiener measure on P (R d ) under I. In order to recall the construction of the Itô map let O(M) denote the orthonormal frame bundle with respect to M, π be the canonical projection O(M) M and H 1,..., H d be the canonical horizontal vector fields. Choose r O(M) such that π(r ) = m. We introduce r γ as the solution to the Stratonovich SDE d r γ (t) = H i (r γ (t)) x i, t [, 1], (7) i=1 r γ () = r, γ = (γ 1,..., γ d ) P (R d ). This defines a.e. a mapping I : P (R d ) P m (M) by I (γ)(t) := π(r γ (t)), γ P (R d ), t [, 1], the Itô map. We also denote by T x M as the tangent space of M at the point x M. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

35 Assumptions Let K C([, )) such that Ric(X, X ) K(r) X 2, X T x M, x B(m, r), r >, (8) where B(m, r) is the geodesic ball at m with radius r. Let ρ denote the Riemannian distance on M and ρ m (x) := ρ(x, m ), x M. We assume that there are constants c 1, c 2, r 1 > such that the following conditions hold, 1 (d 1)K(r) c1 r, r r 1, 2 and { } 1 Ric(X, Y ) p c 2 exp 2 e 1 2c1 ρ m (x) 2 (9) for some p 2 and x M, X, Y T x M, X = Y = 1. We have [ 1 E ] Ric rγ(t) p dt < (1) John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

36 Closability result Proposition If for p specified in (9) we have ĝ ˆϕ L1 (ˆν), ĝ L p 2 (ˆν), (11) then ( ˆD, Ẑ) defined by ˆD( ˆF, Ĝ) := ˆD ˆF, ˆDĜ H ˆϕd ˆν, ˆF, Ĝ Ẑ, is closable on L 2 ( ˆϕˆν). John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

37 ϕ motivation We consider ˆϕ(ˆγ) of the form { 1 ˆϕ(ˆγ) = exp ˆV (ˆγt ), d ˆγ t 1 T ˆγ(t) 2 1 ˆV (ˆγ t ) 2 T ˆγ(t) } dt, ˆγ P m (M). as ˆϕ is then the Radon Nikodym derivative of the diffusion measure corresponding to the general diffusion process on M with generator 1 2 M + ˆV, see [1], with respect to ˆν. We observe that V := r 1 ˆV π r r 1 ˆV I, (12) which says that V V (γ) is an adapted R d -valued process. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

38 Novikov condition in the geometric case The corresponding conditions to (3), (4), i.e. ˆf ˆϕ + ˆf L1 (ˆν) and ˆϕ L1 (ˆν) (13) for all ˆf [ { 1 1 Ẑ, we get Ê exp 2 ˆV }] (ˆγ t ) 2 T ˆγ(t) dt < where Ê denotes expectation taken with respect to ˆν. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

39 Novikov condition in the geometric case The corresponding conditions to (3), (4), i.e. ˆf ˆϕ + ˆf L1 (ˆν) and ˆϕ L1 (ˆν) (13) for all ˆf [ { 1 1 Ẑ, we get Ê exp 2 ˆV }] (ˆγ t ) 2 T ˆγ(t) dt < where Ê denotes expectation taken with respect to ˆν.Thus we have the Novikov condition E [ { 1 exp 2 1 (V (γ)) t 2 R d dt }] <. (14) John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

40 More assumptions In order to obtain quasi-regularity we assume the Ricci curvature to be bounded from below, i.e. there exists some c R, not necessarily non-negative, such that Ric(X, X ) c X 2 X T x M, x M. (15) John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

41 More assumptions In order to obtain quasi-regularity we assume the Ricci curvature to be bounded from below, i.e. there exists some c R, not necessarily non-negative, such that Ric(X, X ) c X 2 X T x M, x M. (15) We also sharpen (1) and assume in addition for some p > 2. [ 1 E ] Ric rγ(t) p dtϕ < (16) John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

42 Quasi-regularity theorem Theorem Assume the following conditions on ˆϕ resp. ϕ, (11), (13), and (16). Let A ε Id for some ε >, and sup c 1,c 2,... {,1} d j=1 p= λip,j ˆϕd ˆν 2 p <. (17) Then Ê( ˆF, ˆF ) = ˆD ˆF, i=1 λ i S i, ˆD ˆF S i H H ˆϕd ˆν, ˆF Ẑ, (18) is quasi-regular in L 2 ( ˆϕˆν). John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

43 Part of proof, flat case E [ Gn,τ 2 ϕ ] = E 2E = 2E 4E sup v {1,...,d} s [,1] sup v {1,...,d} s [,1] sup v {1,...,d} s [,1] [ d sup v=1 s [,1] x v (γ s ) x v (τ s ) ϕ x v (γ s ) ϕ x v ( ) T 1 γ s 2 + C 2 (τ) 2 sup x v (τ s ) 2 v {1,...,d} s [,1] ] M1 v (s) 2 + sup M2 v (s) 2 + C 2 (τ). s [,1] John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

44 Part of proof, flat case cont. t ( ) (T γ) t := γ t bs v (γ) ds, v=1,...,d where b is from (3) such that (5) is satisfied and b v denotes the vth coordinate. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

45 Part of proof, flat case cont. t t ( ) (T γ) t := γ t bs v (γ) ds, v=1,...,d where b is from (3) such that (5) is satisfied and b v denotes the vth coordinate. We have t ( 1 t ) (T 1 γ) v t = γt v + bs v (T 1 γ) ds = 2 γv t + χ {b v s (T 1 γ)>}bs v (T 1 γ) ds ) ( γv t + χ {b v s (T 1 γ) }b v s (T 1 γ) ds =: M v 1 (t) M v 2 (t), John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

46 Part of proof, flat case cont. Using Doob s inequality we get E [ Gn,τ 2 ϕ ] d 16 E [ M1 v (1) 2] d + 16 E [ M2 v (1) 2] + C 2 (τ). (19) v=1 v=1 John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

47 Part of proof, flat case cont. Using Doob s inequality we get E [ Gn,τ 2 ϕ ] d 16 E [ M1 v (1) 2] + 16 v=1 d E [ M2 v (1) 2] + C 2 (τ). (19) v=1 We have E [ [ (1 M1 v (1) 2] 1 = E 2 γv 1 + χ {b v s (T 1 γ)>}bs v (T 1 γ) ds [ ( 1 1 )2 ] 2 E[(γv 1 ) 2 ] + 2E bs v (T 1 γ) ds [ ( ] = 1 1 ) E bs v (γ) ds ϕ(γ) 1 [ 1 ] 2 + 2E bs v (γ) 2 ds ϕ(γ) 1 { 1 [exp 2 + 2E )2 ] } ] bs v (γ) 2 ds ϕ(γ) = 1 [ ] E ϕ. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

48 Different gradients ˆD s ˆF (ˆγ) := k i=1 s s i r 1 I 1 (ˆγ) (s i)( si ˆf )( γ), s [, 1], ˆγ Pm (M), where r is defined in (7). We also introduce k ) ˆD s ˆF (γ) := χ [,si ](s) (T γ ( si si ˆf )( γ), s [, 1], ˆγ P m (M). i=1 In addition as in we define the damped version k ) D s ˆF (γ) := χ [,si ](s) (Q si,st γ ( si si ˆf )( γ), s [, 1], ˆγ P m (M), (2) i=1 where Q denotes the adjoint of Q s,t : R d R d, which is defined by or equivalently dq s,t ds = 1 2 Ric r γ(s)q s,t, Q t,t = Id Tm, t s, dq s,t dt = 1 2 Q s,tric rγ(t), Q s,s = Id Tm, t s. (21) John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

49 References M. Capitaine, E. P. Hsu, and M. Ledoux. Martingale representation and a simple proof of logarithmic sobolev inequalities on path spaces. Electron. Comm. Probab., 2:71 81, C. Houdré and N. Privault. A concentration inequality on Riemannian path space. In Stochastic inequalities and applications, volume 56 of Progr. Probab., pages Birkhäuser, Basel, 23. E. P. Hsu and C. Ouyang. Quasi-invariance of the wiener measure on the path space over a complete riemannian manifold. J. Funct. Anal., 257(5): , 29. F.-Y. Wang and B. Wu. Quasi-regular dirichlet forms on riemannian path and loop spaces. Forum Math. Volume, 2(6): , 28. F.-Y. Wang and B. Wu. Quasi-regular dirichlet forms on free riemannian path spaces. John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3

50 Thank you for listening John Karlsson (Linköping University) Infinite dimensional stochastic processes CRM Barcelona, July 7, / 3