Stochastic analysis for Markov processes

Transcription

1 Colloquium Stochastic Analysis, Leibniz University Hannover Jan. 29, 2015 Universität Bielefeld

2 1 Markov processes: trivia. 2 Stochastic analysis for additive functionals. 3 Applications to geometry.

3 Markov processes X locally compact separable metric space. A stochastic process Y = (Y t ) t 0 is Markov process with state space X if (very loosely speaking!) there is a family (P x ) x X of p.m. s on (Ω, F) such that x P x (Y t A) is a Borel function for all Borel sets A X and all t 0, with F t := σ(y s : s t) we have P x [Y t+s A F t ] = P Y t [Y s A] for all s, t 0 and A X Borel ( process forgets past, given present )

4 Example d-dim. Brownian motion (B t ) t 0 (with varying starting points) is a Markov process with state space R d.

5 Consider suitable volume measure m on X ( speed measure ). Y is m-symmetric if E m [f (Y t )g(y 0 )] = E m [f (Y 0 )g(y t )] for all t > 0 and bounded Borel f, g. Here P m = X Px m(dx) and E m expectation w.r.t. P m.

6 There is a probability kernel P t (x, dy) such that P x (Y t A) = P t (x, dy). By m-symmetry A P t f (x) := E x [f (Y t )] defines a strongly continuous Markovian semigroup (P t ) t 0 of symmetric operators on L 2 (X, m) with generator 1 Lf := lim t 0 t (P tf f ), f dom L. L non positive definite self-adjoint on L 2 (X, m).

7 Example For d-dim. Brownian motion (B t ) t 0 have P t f (x) = p(t, x y)f (y)dy R d with symmetric on L 2 (R d ). Generator is p(t, x) = 1 ) exp ( x 2, 2πt 2t 1 2 = 1 2 f 2 x 2 i i (Friedrichs extension ( 1 2, H2 (R d )).

8 Connect with martingale theory: Theorem (Doob, Kakutani, Dynkin) If f dom L (and nice) then for q.e. x X f (Y t ) f (Y 0 ) t is a P x -martingale (w.r.t. natural filtration ). 0 (Lf )(Y s )ds

9 Example If (B t ) t 0 Brownian motion on R d and f is C 2 then Itô formula holds, f (B t ) f (B 0 ) 1 2 t 0 ( f )(B s )ds = i t 0 f x i (B s )db i s. If h harmonic then h(b t ) forms martingale for any P x.

10 Energy and additive functionals Relax hypotheses by using energy forms. Consider the unique symmetric positive definite bilinear form (Q, dom Q) on L 2 (X, m) such that Q(f, g) := (Lf, g) L2 (X,m), f dom L, g dom Q. (Dirichlet form). Examples (B t ) t 0 d-dim Brownian motion, then Q(f, g) = 1 f g dx, 2 R d f, g H 1 (R d ) dom = H 2 (R d ).

11 Theorem (Fukushima) If f dom Q and nice, then M [f ] t = f (Y t ) f (Y 0 ) N [f ] t (uniquely) where (M [f ] t ) t 0 a continuous martingale additive functional of Y of finite energy, and (N [f ] t ) t 0 an continuous additive functional of Y of zero energy. This is sth. like a semimartingale decomposition. Problem: family (P x ) x X of p.m. s.

12 Additive functionals: Examples If B Brownian motion on R d then A t = t 0 g(b s )ds is a continuous additive functional of B, additivity property is t+s 0 g(b r )dr = s 0 t g(b r )dr + g(b r+s )dr a.s. 0

13 Space of continuous AF s of zero energy ( analytically nice ): N c := {N : N finite continuous AF of Y with e(n) = 0 where ( finite quadratic variation part ) and such that E x ( N t ) < + q.e. for each t > 0}, 1 e(m) = lim t 0 2t Em (Mt 2 ).

14 Space of martingale additive functionals of finite energy ( probabilistically nice ): M = {M : M AF of Y with e(m) < such that } for q.e. x X, E x (Mt 2 ) < and E x (M t ) = 0, t > 0, The space ( M, e) is Hilbert.

15 To each M M assign energy measure µ M... Revuz measure of its sharp bracket M : For q.e. x X, M 2 M is a P x -martingale (Doob-Meyer version). For h 0 Borel and f dom Q (nice) have ( t ) E hm f (Y s )d M s = 0 t 0 X E x h(y s )f (x) µ M (dx)ds, t > 0. ( Fubini with trading strange scaling (time change) between time and space )

16 Examples If B is BM on R d and µ(dx) = g(x)dx then µ is Revuz measure of Examples A t = t 0 g(b s )ds. If B is BM on R and δ y Dirac at y, then up to a constant, δ y is the Revuz measure of Brownian local time L(t, y), t 0 1 E (B s )ds = 2 L(t, y)dy, E R Borel. E

17 Stochastic integrals For f L 2 (X, µ M ) can define the stochastic integral f M M of f with respect to M M by e(f M, N) = 1 fdµ 2 M,N, N M. The integral f M is an L 2 -limit of sums f (Y ti )(M ti+1 M ti ) i (Itô type). Not known how to use general integrands. X

18 Example If B = (B 1,..., B d ) is the d-dim. Brownian motion, seen as Markov process, then { d } M = f i B i : f i L 2 (R d ), i = 1,..., d i=1 and ( d ) e f i B i = 1 2 i=1 d f i 2 L 2 (R d ). i=1

19 Definition (Motoo/Watanabe, Hino) The martingale dimension of (Y t ) t 0 is the smallest natural number p such that there exist M (1),..., M (p) M allowing the representation M t = p (h i M (i) ) t, t > 0, P x -a.e. for q.e. x X, i=1 with suitable h i L 2 (X, µ M (i) ) for every M M. If no such p exists, we define the martingale dimension to be infinity.

20 Examples Martingale dimension of d-dim. Brownian motion is d. Additive functional version of martingale representation. Exact relation between the formulations is not yet understood.

21 (B t ) t 0 one dim. Brownian motion ( on a p. space (Ω, F, P), ) F t := σ(b s : 0 s t), F := σ t 0 F t. Lemma For all random variables F L 2 (Ω, F, P) there exists a unique predictable process H which is in L 2 and satisfies F = EF + 0 H s db s P a.s. ( space of stochastic integrals is large ).

22 Now based on d-dim. Brownian motion (B 1,..., B d ): Theorem Let (M t ) t 0 be an d-dim. L 2 -integrable (F t ) t 0 -martingale. Then there are a constant C and predictable processes H i, i = 1,..., d in L 2 such that d t M t = C + HsdB i s i a.s. i=1 Think of d as degree of freedom for heat particle. 0

23 Geometry of rough spaces Lungs.

24 Artificial fern.

25 Sponge.

26 Menger sponge.

27 Refraction patterns in Laser optics.

28 Hofstadter Butterfly (energy spectra, magnetic field on square lattice).

29 Hofstadter Butterfly observed on Graphene structure.

30 Interest: Geometry, analysis, stochastic processes, math. physics on rough spaces (no rectifiability or curvature dimension bounds, fractals ) Study microstructure... complement homogenization. Problem: Credo: Classical differentiation unavailable. Diffusion processes exist and can be used. Dimension issues (topological, Hausdorff, martingale,...) Diffusion does not need smoothness.

31 Some applications / motivations: Waveguides for optical high frequency signals. Fractal antennas Fractal structuring : Separating layers between polymer films. Ultra light weight materials. Networks at different scales. Fractal microcavities. Nanotubes. Geometric learning and pattern recognition. Space-time scaling in models for quantum gravity.

32 Sierpinski carpet Barlow/Bass 89 (existence of Brownian motion), Barlow/Bass/Kumagai/Teplyaev 10 (uniqueness).

33 Honeycomb structure (stable ultra light weight material, US patent).

34 Pyramid structure with huge surface.

35 Sierpinski gasket SG Barlow/Perkins 88, Kigami 89 (ex. and uniqueness of Brownian motion).

36 d H = log 3 log 2 Hausdorff dimension of SG d w = log 5 log 2 > 2 walk index, c 1 t 2/dw E x Y t Y 0 2 c 2 t 2/dw ( particle moves slower than normal ) d S = 2d H /d w < 2 spectral dimension, short time exponent diffusion is sub-gaussian, i.e. ( p(t, x, y) ct ds/2 exp c ( ) dr (x, y) dw 1/(1 dw ) ). log-scale fluctuations in on-diagonal behaviour t ds/2 p(t, x, x) (Kajino) t

37 Construct energy functional as the (rescaled) limit E(f ) = E(f ) = lim n ( 5 3 ) n f (x) 2 dx p,q V n, q p (f (p) f (q)) 2 of discrete energy forms on approximating graphs (Kigami 89, 93, Kusuoka 93)

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39 Get a space F of functions on SG with finite energy, i.e. E : F [0, + ). Simultaneously get a (resistance) metric d R on SG so that F C(SG) (Sobolev embedding theorem).

40 Construction is purely combinatorial. With any reasonable finite Borel measure µ on SG the pair (E, F) becomes a Dirichlet form on L 2 (SG, µ). Integration by parts also yields Laplacian (generator) µ for (speed) measure µ, E(f, g) = f µ g dµ. SG ( Second derivative on fractals ) Fukushima s theory yields associated diffusion ( Brownian motion on SG )

41 Analytic counterpart Recall P t f (x) = E x [f (Y t )], where (Y t ) t 0 diffusion on X. Then 1 Q(f, g) := lim t 0 2t (f P tf, g) L2 (X,m). (Q, dom Q) strongly local regular symmetric Dirichlet form on L 2 (X, m). The core C := C c (X) dom Q is an algebra.

42 On C C consider the nonnegative def. symmetric bilinear form a b, c d H := Q(bda, c) + Q(a, bdc) Q(ac, bd). Factoring out zero seminorm elements yields Hilbert space H of differential 1-forms / vector fields. (Mokobodzki, LeJan, Nakao, Lyons/Zhang, Eberle, Cipriani/Sauvageot, etc.) Close to algebra and NCG.

43 H can be given module structure the operator : C H with f := f 1 is a bounded derivation ( f is H-class universal derivation / Kähler differential of f ). Examples M compact Riemannian manifold, (Y t ) t 0 Brownian motion on M, Q(f, g) = df, dg T M dvol, f, g H1 (M), M dvol Riemannian volume, d exterior derivative. Then H = L 2 (M, dvol, T M) and coincides with d.

44 Theorem There are a suitable measure ν and suitable Hilbert spaces H x such that H may be written as direct integral, H = X H x ν(dx). The fibers H x may be regarded as (co)tangent spaces at x to X. Examples Manifold case: H x = Tx M for dvol-a.e. x.

45 Theorem The spaces H and M are isometrically isomorphic under g f g M [f ]. (Nakao: manifolds, H./Teplyaev/Röckner: fractals) Theorem (Hino) The martingale dimension of (Y t ) t 0 equals ess sup x X dim H x. ( maximal degree of freedom for diffusing particle is essentially given by maximal tangent space dimension )

46 Examples The (harmonic) Sierpinski gasket has tangent spaces of dimension one a.e.

47 Play with this correspondence: gradient f... martingale AF M [f ] divergence v... Revuz measure (density) of Nakao functional vector field g f... stochastic integral g M [f ] etc.

48 Some results Theorem with Stratonovich integral P a,v t f (x) := E x [e i Y ([0,t]) a t 0 v(ys)ds f (Y t )] Y ([0,t]) a := Θ(a) + is semigroup for magnetic Hamiltonian Θ : H M Nakao isomorphism. t 0 ( a)(y s )ds H a,v = ( + ia) ( + ia) + v. ( Feynman-Kac-Itô, H. 14)

49 Theorem Hodge theorem in topo dim one: H = Im (locally) harmonic forms. (Ionescu/Rogers/Teplyaev 11, H./Teplyaev 12) Theorem Harmonic forms give Čech cohomology (Ionescu/Rogers/Teplyaev 11, H./Teplyaev 12)

50 Theorem If topo dimension is one (but Hausdorff dim ), Navier-Stokes system reduces to Euler equation. (H./Teplyaev 12)

51 Q k = log((2k + 1)2 1) = log(4k2 + 4k) < 2 Theorem log(2k + 1) log(2k + 1) orsifregular topo dimension in that dimension. is one, then either martingale dimension is one or ting exterior point derivation for our investigations is not closable. was the following well-known fact. n(h./teplyaev 1.4. For each 15) k, (unprecedented the carpet S 1/(2k+1) in diff., equipped geo) with Euclidean metric and its dimension Q k, does not support any Poincaré inequality. Figure 1. S 1/3 Figure 2. S (1/3,1/5,1/7,...)

52 THANK YOU.