Quantum Wiener chaos expansion

Transcription

1 Lancaster University, United Kingdom (joint work with Martin Lindsay) Workshop on Operator Spaces, Locally Compact Quantum Groups and Amenability Toronto, 29 th May 2014

2 Integration - product formulae Classical integration by parts Let x t = t 0 α(s)ds, y t = t 0 β(s)ds. Then x t y t = = t (α(s)y s + x s β(s))ds 0 t s 0 0 (α(s)β(r) + α(r)β(s))drds

3 Integration - product formulae Wiener integration by parts Let X t = t 0 f (s)dw s, Y t = t 0 g(s)dw s. Then x t y t = = t (α(s)y s + x s β(s))ds 0 t s 0 0 (α(s)β(r) + α(r)β(s))drds

4 Integration - product formulae Wiener integration by parts Let X t = t 0 f (s)dw s, Y t = t 0 g(s)dw s. Then X t Y t = = t (f (s)y s + X s g(s))dw s + 0 t s 0 0 t 0 f (s)g(s)ds (f (s)g(r) + f (r)g(s))dw r dw s + t 0 f (s)g(s)ds

5 Integration - product formulae Quantum stochastic integration by parts Let X t = t 0 F (s)dλ s, Y t = t 0 G(s)dΛ s. Then X t Y t = = t (f (s)y s + X s g(s))dw s + 0 t s 0 0 t 0 f (s)g(s)ds (f (s)g(r) + f (r)g(s))dw r dw s + t 0 f (s)g(s)ds

6 Integration - product formulae Quantum stochastic integration by parts Let X t = t 0 F (s)dλ s, Y t = t 0 G(s)dΛ s. Then X t Y t = = t 0 t s 0 (F s Y s + X s G s + F s G s )dλ s t (F r G s + F s G r )dλ r dλ s + F s G s dλ s. 0 0

7 Motivations Introduction 1 Fock space provides a universal arena for stochastic calculus.

8 Motivations Introduction 1 Fock space provides a universal arena for stochastic calculus. 2 Duality transforms induce algebraic structure on Fock space.

9 Motivations Introduction 1 Fock space provides a universal arena for stochastic calculus. 2 Duality transforms induce algebraic structure on Fock space. 3 Products of stochastic integrals are expressible in closed-form formulas.

10 Motivations Introduction 1 Fock space provides a universal arena for stochastic calculus. 2 Duality transforms induce algebraic structure on Fock space. 3 Products of stochastic integrals are expressible in closed-form formulas. 4 Known classical examples are particular cases of a general quantum picture.

11 Multiple Wiener integrals Let (W t ) t 0 be the canonical one-dimensional Wiener process on (Ω, F, P).

12 Multiple Wiener integrals Let (W t ) t 0 be the canonical one-dimensional Wiener process on (Ω, F, P). One can form multiple Wiener integrals: I n (f ) = f (t 1,, t n )dw t1...dw tn L 2 (Ω, F, P) n f L 2 ( n ), n = {(t 1,, t n ) [0, ) n : t 1... t n }.

13 Multiple Wiener integrals Let (W t ) t 0 be the canonical one-dimensional Wiener process on (Ω, F, P). One can form multiple Wiener integrals: I n (f ) = f (t 1,, t n )dw t1...dw tn L 2 (Ω, F, P) n f L 2 ( n ), n = {(t 1,, t n ) [0, ) n : t 1... t n }. Theorem (Wiener-Segal-Itô) L 2 (Ω, F, P) = I n (L 2 ( n )). n 0

14 Multiple Wiener integrals Let (W t ) t 0 be the canonical one-dimensional Wiener process on (Ω, F, P). One can form multiple Wiener integrals: I n (f ) = f (t 1,, t n )dw t1...dw tn L 2 (Ω, F, P) n f L 2 ( n ), n = {(t 1,, t n ) [0, ) n : t 1... t n }. Theorem (Wiener-Segal-Itô) L 2 (Ω, F, P) = I n (L 2 ( n )). n 0 Also: compensated Poisson process, certain Azéma martingales.

15 Noncommutative probability Probability Quantum probability

16 Noncommutative probability Probability L (Ω, F, P) Quantum probability

17 Noncommutative probability Probability Quantum probability L (Ω, F, P) (A, ω), A v.n.a., ω normal state

18 Noncommutative probability Probability Quantum probability L (Ω, F, P) (A, ω), A v.n.a., ω normal state 1 A, A F

19 Noncommutative probability Probability Quantum probability L (Ω, F, P) (A, ω), A v.n.a., ω normal state 1 A, A F p = p = p 2 A

20 Noncommutative probability Probability Quantum probability L (Ω, F, P) (A, ω), A v.n.a., ω normal state 1 A, A F p = p = p 2 A f : Ω X r.v. F : B A unital homomorphism

21 Noncommutative probability Probability Quantum probability L (Ω, F, P) (A, ω), A v.n.a., ω normal state 1 A, A F p = p = p 2 A f : Ω X r.v. F : B A unital homomorphism P f 1 ω F.

22 Fock space Introduction Fix Hilbert space k. Set K = L 2 (R + ; k), k n = k... k (k 0 = C) and Symmetric Fock space over K: K n = Lin{f n : f K}. F k = Γ (K) = n 0 K n.

23 Fock space Introduction Fix Hilbert space k. Set K = L 2 (R + ; k), k n = k... k (k 0 = C) and Symmetric Fock space over K: K n = Lin{f n : f K}. F k = Γ (K) = n 0 K n. K and F decompose as follows: K = K [0,t[ K [t, [, F k = F k [0,t[ F k [t, [, where K I := L 2 (I ; k) and F k I := Γ (K I ) for I R +.

24 Exponential vectors ε(f ) := (1, f, f 2 2,, f n n!,...).

25 Exponential vectors ε(f ) := (1, f, f 2 2,, f n n!,...). Exp := Lin{ε(f ): f L 2 (R + ; k)}.

26 Exponential vectors ε(f ) := (1, f, f 2 2,, f n n!,...). Exp := Lin{ε(f ): f L 2 (R + ; k)}. Ω := ε(0) = (1, 0, 0,...).

27 Exponential vectors ε(f ) := (1, f, f 2 2,, f n n!,...). Exp := Lin{ε(f ): f L 2 (R + ; k)}. Ω := ε(0) = (1, 0, 0,...). ω Ω (A) = Ω, AΩ.

28 Topological version of Schürmann Reconstruction Theorem Theorem (Das Lindsay 2012, Lindsay Skalski 2004) Every quantum Lévy process is (equivalent to) a Fock space quantum Lévy process, i.e. a process (j s,t : A (B(F k ), ω Ω )) 0 s t for some Hilbert space k.

29 Topological version of Schürmann Reconstruction Theorem Theorem (Das Lindsay 2012, Lindsay Skalski 2004) Every quantum Lévy process is (equivalent to) a Fock space quantum Lévy process, i.e. a process (j s,t : A (B(F k ), ω Ω )) 0 s t for some Hilbert space k. Moreover, this process is expressible in terms of multiple quantum Wiener integrals.

30 Guichardet space 1: Definition Definition Γ = Fin(R + ); Γ s,t = Fin([s, t[)

31 Guichardet space 1: Definition Definition Γ = Fin(R + ); Γ s,t = Fin([s, t[) {s 1 <... < s n } (s 1,, s n ) R n +;

32 Guichardet space 1: Definition Definition Γ = Fin(R + ); Γ s,t = Fin([s, t[) {s 1 <... < s n } (s 1,, s n ) R n +; µ({ }) = 1.

33 Guichardet space 1: Definition Definition Γ = Fin(R + ); Γ s,t = Fin([s, t[) {s 1 <... < s n } (s 1,, s n ) R n +; µ({ }) = 1. µ(γ 0,t ) = e t.

34 Guichardet space 1: Definition Definition Γ = Fin(R + ); Γ s,t = Fin([s, t[) {s 1 <... < s n } (s 1,, s n ) R n +; µ({ }) = 1. µ(γ 0,t ) = e t. We denote the sets by Greek letters - α, β, γ.

35 Guichardet space 1: Definition Definition Γ = Fin(R + ); Γ s,t = Fin([s, t[) {s 1 <... < s n } (s 1,, s n ) R n +; µ({ }) = 1. µ(γ 0,t ) = e t. We denote the sets by Greek letters - α, β, γ. Definition F k = {F L 2 (Γ ; k n ) : σ Γ F (σ) k #σ }. n 0

36 Guichardet space 2: Explanation Let x be a vector from the Fock space over L 2 (R + ; k) = K. Then x (C, K, K 2,..., )

37 Guichardet space 2: Explanation Let x be a vector from the Fock space over L 2 (R + ; k) = K. Then x(σ) (C, k, k 2,..., )

38 Guichardet space 2: Explanation Let x be a vector from the Fock space over L 2 (R + ; k) = K. Then x(σ) (C, k, k 2,..., ) σ

39 Guichardet space 2: Explanation Let x be a vector from the Fock space over L 2 (R + ; k) = K. Then x(σ) (C, k, k 2,..., ) σ = {s 1 <... < s n }

40 Guichardet space 2: Explanation Let x be a vector from the Fock space over L 2 (R + ; k) = K. Then x(σ) (C, k, k 2,..., k n,...) σ = {s 1 <... < s n }

41 Guichardet space 2: Explanation Let x be a vector from the Fock space over L 2 (R + ; k) = K. Then x(σ) (C, k, k 2,..., k n,...) σ = {s 1 <... < s n } f L 2 (R + ; k) : π f ( ) = 1 C, π f (σ) = s σ f (s), π f ε(f ).

42 Guichardet space 3: quantum Wiener integrands Let k := C k.

43 Guichardet space 3: quantum Wiener integrands Let k := C k. For f L 2 (R + ; k) let f (s) = ( 1 f (s)) k.

44 Guichardet space 3: quantum Wiener integrands Let k := C k. For f L 2 (R + ; k) let f (s) = ( 1 f (s)) k. Consider F: Γ n N B( k n ) such that F(σ) B( k #σ ).

45 Guichardet space 3: quantum Wiener integrands Let k := C k. For f L 2 (R + ; k) let f (s) = ( 1 f (s)) k. Consider F: Γ n N B( k n ) such that F(σ) B( k #σ ). Example For F : R + B( k) we can define π F : Γ n N B( k n ) by π F ({s 1 <... < s n }) = F s1... F sn.

46 Guichardet space 4: quantum Wiener integrals Implicit definition: Multiple quantum Wiener integral π f, Λ N s,t(f)π g = e f,g Γ s,t dσ π f (σ), F(σ)πĝ (σ). Λ N t := Λ N 0,t = n 0 Λ n t.

47 Guichardet space 4: quantum Wiener integrals Implicit definition: Multiple quantum Wiener integral π f, Λ N s,t(f)π g = e f,g Γ s,t dσ π f (σ), F(σ)πĝ (σ). Λ N t := Λ N 0,t = n 0 Λ n t. Bi-adaptedness property: Λ N s,t( ) = I F[0,s[ ( ) I F[t, [.

48 Guichardet space 4: quantum Wiener integrals Implicit definition: Multiple quantum Wiener integral π f, Λ N s,t(f)π g = e f,g Γ s,t dσ π f (σ), F(σ)πĝ (σ). Λ N t := Λ N 0,t = n 0 Λ n t. Bi-adaptedness property: Λ N s,t( ) = I F[0,s[ ( ) I F[t, [. Evolution property: Λ N r,s(π F )Λ N s,t(π F ) = Λ N r,t(π F ), r < s < t.

49 Well-definedness of the integral Proposition ( ) ( ) K M B(C) B(k; C) Let F = : R L N + B( k) = have the B(C; k) B(k) property that K L 1 loc (R +), L, M L 2 loc (R +), N L loc (R +). Then Dom(Λ N t (π F )) Exp.

50 Well-definedness of the integral Proposition ( ) ( ) K M B(C) B(k; C) Let F = : R L N + B( k) = have the B(C; k) B(k) property that K L 1 loc (R +), L, M L 2 loc (R +), N L loc (R +). Then Dom(Λ N t (π F )) Exp. Proposition Let F L 1,2,2,. Then Dom(Λ N t (F)) Exp.

51 Quantum Wiener convolution Definition For F, G L 1,2,2,, define F G: Γ n N B( k n ) by F G(σ) = F(α; σ) (α β; σ)g(β; σ). α β=σ

52 Quantum Wiener convolution Definition For F, G L 1,2,2,, define F G: Γ n N B( k n ) by F G(σ) = F(α; σ) (α β; σ)g(β; σ). α β=σ

53 Quantum Wiener convolution Definition For F, G L 1,2,2,, define F G: Γ n N B( k n ) by F G(σ) = F(α; σ) (α β; σ)g(β; σ). α β=σ

54 Placement notation Let k n N, σ = {s 1,, s n }, α = {s i1,, s ik }. X = X 1... X k, X i B( k). n X (α; σ)(u 1... u n ) = Y i u i, where Y i = i=1 { X j i = i j, j {1,, k}; I i / {i 1,, i k }.

55 Placement notation Let k n N, σ = {s 1,, s n }, α = {s i1,, s ik }. X = X 1... X k, X i B( k). n X (α; σ)(u 1... u n ) = Y i u i, where Y i = i=1 { X j i = i j, j {1,, k}; I i / {i 1,, i k }. X (α; σ) = X i1 i k.

56 Placement notation Let k n N, σ = {s 1,, s n }, α = {s i1,, s ik }. X = X 1... X k, X i B( k). n X (α; σ)(u 1... u n ) = Y i u i, where Y i = i=1 { X j i = i j, j {1,, k}; I i / {i 1,, i k }. X (α; σ) ( = X i1 i ) k. 0 0 =. 0 I k

57 Placement notation Let k n N, σ = {s 1,, s n }, α = {s i1,, s ik }. X = X 1... X k, X i B( k). n X (α; σ)(u 1... u n ) = Y i u i, where Y i = F G(σ) = i=1 { X j i = i j, j {1,, k}; I i / {i 1,, i k }. X (α; σ) ( = X i1 i ) k. 0 0 =. 0 I k F(α; σ) (α β; σ)g(β; σ). α β=σ

58 Quantum Wiener product formula Theorem (MJ-Lindsay) Let F, G L 1,2,2,. Then F G L 1,2,2,

59 Quantum Wiener product formula Theorem (MJ-Lindsay) Let F, G L 1,2,2,. Then F G L 1,2,2, and Λ N (F)Λ N (G) = Λ N (F G) on Exp.

60 Applications Introduction 1 Universal picture of various stochastic products: k = C,

61 Applications Introduction 1 Universal picture of various stochastic products: k = C, F = L 2 (Γ ), f, g F,

62 Applications Introduction 1 Universal picture of various stochastic products: k = C, F = L 2 (Γ ), f, g F, (f W g)(σ) = dωf (α ω)g((σ \ α) ω). α σ Γ

63 Applications Introduction 1 Universal picture of various stochastic products: k = C, F = L 2 (Γ ), f, g F, (f W g)(σ) = dωf (α ω)g((σ \ α) ω). α σ Γ (f P g)(σ) = α β=σ Γ dωf (α ω)g(β ω).

64 Applications Introduction 1 Universal picture of various stochastic products: k = C, F = L 2 (Γ ), f, g F, (f W g)(σ) = dωf (α ω)g((σ \ α) ω). α σ Γ (f P g)(σ) = α β=σ Γ 2 Lie-Trotter product formulae. dωf (α ω)g(β ω).

65 References Introduction M.Jurczyński, J.M. Lindsay, Quantum Wiener chaos, in preparation. U. Franz, Lévy processes on quantum groups and dual groups, Lect. Notes Math. 1866, (2006), Springer-Verlag Berlin Heidelberg 2006 A. Guichardet, Symmetric Hilbert spaces and Related Topics, Lect. Notes Math., J.M. Lindsay, H. Maassen, Duality transform as -algebraic isomorphism, Quantum Probability and Applications V, Lect. Notes Math., Springer-Verlag Berlin Heidelberg 1988 J.M. Lindsay, A. Skalski, Quantum stochastic convolution cocycles I, Ann. de l Institut H. Poincare, Vol 41, Issue 3, J.M. Lindsay, S.J. Wills, Homomorphic Feller Cocycles on a C -algebra, JLMS (2) 68 (2003) I.E. Segal, Tensor algebras over Hilbert spaces. I, Trans. AMS, Vol. 81, No. 1 (1956). A. Skalski, Quantum Lévy processes on topological quantum groups, Lecture series, 3 ILJU School of Mathematics, Gyeong-Ju (Korea), N. Wiener, The homogeneous chaos, American Journal of Mathematics, Vol. 60, No. 4, 1938.