On rough PDEs. Samy Tindel. University of Nancy. Rough Paths Analysis and Related Topics - Nagoya 2012

Transcription

1 On rough PDEs Samy Tindel University of Nancy Rough Paths Analysis and Related Topics - Nagoya 2012 Joint work with: Aurélien Deya and Massimiliano Gubinelli Samy T. (Nancy) On rough PDEs Nagoya / 23

2 Sketch of the talk 1 Introduction Pathwise type stochastic PDEs Main aim 2 Description of the results Abstract setting Applications Heuristic computations in the Young case Samy T. (Nancy) On rough PDEs Nagoya / 23

3 Sketch of the talk 1 Introduction Pathwise type stochastic PDEs Main aim 2 Description of the results Abstract setting Applications Heuristic computations in the Young case Samy T. (Nancy) On rough PDEs Nagoya / 23

4 Sketch of the talk 1 Introduction Pathwise type stochastic PDEs Main aim 2 Description of the results Abstract setting Applications Heuristic computations in the Young case Samy T. (Nancy) On rough PDEs Nagoya / 23

5 Equation under consideration Equation: dy t = Ay t dt + f (y t )dx t, with y 0 = ψ t [0, T ], y t B, with B of the form L p A = Laplace operator, with semi-group (S t ) t 0 Generalization to divergence type operators f from B to an operator space x general noise, γ-hölder continuous, γ > 1/3 Mild formulation: y t = S t ψ + t 0 S ts f (y s )dx s, with S ts := S t s Motivations: Natural generalization of SPDEs, continuity results Samy T. (Nancy) On rough PDEs Nagoya / 23

6 Methods of resolution (1) Brownian case Peszat-Zabczyk, Dalang (90s) Well understood by probabilistic methods Additive noise, x inf-dim fbm, for any H Tindel-Tudor-Viens ( 03) Wiener integrals estimates for fbm Non-linear case, inf-dim fbm, H > 1/2 Maslowski-Nualart ( 03), smooth noise in space Fractional integrals Linear and nonlinear cases, x finite-dim Caruana-Friz ( 08-09), Diehl-Friz ( 11) Abstract setting of rough paths, viscosity solutions Samy T. (Nancy) On rough PDEs Nagoya / 23

7 Methods of resolution (2) Non-linear case, x γ-hölder path with γ > 1/2 Non-smooth noise in space Young integration in infinite dimension Local solution, due to the fact that f is only locally Lipschitz Lejay-Gubinelli-Tindel ( 06) Wave equation, d = 1, x space-time noise with γ > 1/2 Young integration in the plane Quer-Tindel ( 07) Genuine rough paths setting for SPDEs Applications: global solution for fbm, H > 5/6 Brownian case, specific cases of f, infinite dimensional noise Gubinelli-Tindel ( 10) Samy T. (Nancy) On rough PDEs Nagoya / 23

8 Sketch of the talk 1 Introduction Pathwise type stochastic PDEs Main aim 2 Description of the results Abstract setting Applications Heuristic computations in the Young case Samy T. (Nancy) On rough PDEs Nagoya / 23

9 General aim Existence and uniqueness results in the mild sense for dy t = Ay t dt + f (y t )dx t (1) y t function in space, Hölder continuous in time A Laplace operator, f non-linear coefficient x t finite dimensional, γ-hölder continuous in time, γ > 1/3 Application: x fbm, with H > 1/3 Bibliographical note: Local solution obtained in Deya-Gubinelli-T. (PTRF, to appear) Global solution obtained by Deya (Elec. J. Probability) Samy T. (Nancy) On rough PDEs Nagoya / 23

10 Sketch of the talk 1 Introduction Pathwise type stochastic PDEs Main aim 2 Description of the results Abstract setting Applications Heuristic computations in the Young case Samy T. (Nancy) On rough PDEs Nagoya / 23

11 Sketch of the talk 1 Introduction Pathwise type stochastic PDEs Main aim 2 Description of the results Abstract setting Applications Heuristic computations in the Young case Samy T. (Nancy) On rough PDEs Nagoya / 23

12 The rough paths black box for SDEs Hypothesis: x C γ (R d ) with γ > 1/3 x allows to define a Levy area x 2 C 2γ (R d d ) dx dx Coefficient σ C 3 b Main rough paths theorem: Let y be the solution to y t = a + t 0 σ(y s ) dx s. Then (Lyons-Qian, Friz-Victoir, Gubinelli) F : R n C γ (R d ) C 2γ (R d d ) C γ (R n ), is a continuous map (a, x, x 2 ) y Levy area Rough paths machinery Solution to RDEs Smooth coefficients Samy T. (Nancy) On rough PDEs Nagoya / 23

13 State space for the solution (SPDEs) L p space: B p := L p (R n ) Fractional Sobolev spaces: for α [0, 1/2), B α,p := W 2α,p (R n ) = [Id ] α (L p (R n )) Action of the heat semigroup: Contraction: S t contraction on B α,p Regularization: S t ϕ Bα,p c α t α ϕ Bp. Remark: We work with B α,p with large p Because we want this space to be an algebra. Samy T. (Nancy) On rough PDEs Nagoya / 23

14 A (slightly) unusual formulation We solve equation (1) under the form: N t y t = S t ψ + S ts dxs i f i (y s ), (2) i=1 0 f i function from B p to B p x i scalar noise Formulation (2) fits better to rough path type expansions Example of nonlinear term: [f (ϕ)](ξ) = σ(ξ, ϕ(ξ)), where σ : R n R R rapidly decreasing function in ξ. Samy T. (Nancy) On rough PDEs Nagoya / 23

15 Operator valued Levy area First order integrals: a us = S us Id X x,i ts (ϕ) = X xa,i ts (ϕ, ψ) = t s t s S tu (ϕ) dx i u S tu [a us (ϕ) ψ] dx i u Second order integral: δx us = x u x s t Xts xx,ij (ϕ) = S tu (ϕ) δxus j dxu i s More specifically: if g := heat kernel [ X xx,ij ts (ϕ) ] t ( ) (ξ) = g t u (ξ η)ϕ(η) dη δx j s R n us dxu i Samy T. (Nancy) On rough PDEs Nagoya / 23

16 General result (loose formulation) Theorem Assume x allows to define X x, X xa and X xx (operator valued on B γ,p ) X x C γ and X xa, X xx C 2γ γ > 1/3 f and initial condition ψ regular enough (ψ B γ,p ) Then the equation y t = S t ψ + t has a unique global solution y on [0, T ] 0 S ts f (y s ) dx s y is a continuous function of X x, X xa and X xx Samy T. (Nancy) On rough PDEs Nagoya / 23

17 Sketch of the talk 1 Introduction Pathwise type stochastic PDEs Main aim 2 Description of the results Abstract setting Applications Heuristic computations in the Young case Samy T. (Nancy) On rough PDEs Nagoya / 23

18 Examples of application Space-time dependence: {σ j ; j N} collection of rapidly decreasing smooth functions Generic form of the noise: x t = N j=1 σ j B j t Application in the Young setting: B j fbm with H > 1/2 Application in the rough setting: B j fbm with H > 1/3 Equation which can be solved: N t y t (ξ) = y t (ξ) + σ j (ξ) f (y t (ξ)) dbt j j=1 Samy T. (Nancy) On rough PDEs Nagoya / 23

19 Sketch of the talk 1 Introduction Pathwise type stochastic PDEs Main aim 2 Description of the results Abstract setting Applications Heuristic computations in the Young case Samy T. (Nancy) On rough PDEs Nagoya / 23

20 Setting Notational simplification: x C γ with γ > 1/2, 1-dimensional [f (y)](ξ) = σ(ξ, y(ξ)). Equation we wish to solve: y t = S t ψ + t 0 S ts dx s f (y s ) Hypothesis: The solution y t exists in a space C γ ([0, T ]; B p ), with γ > 1/2 Main step: define the integral t 0 S ts dx s f (y s ) fixed point argument Samy T. (Nancy) On rough PDEs Nagoya / 23

21 Formal computations One way to catch the regularity of the solution y: t (ˆδy) ts := y t S ts y s = S tu dx u f (y u ) s ( t ) t = S tu dx u f (y s ) + S tu dx u δ(f (y)) us Definition of the terms: s = X x ts f (y s ) + y ts X x ts f (y s ) well-defined as long as X x is well-defined as an operator acting on B p y ts defined as a Young integral if γ > 1/2 with Hölder regularity > 1 s Samy T. (Nancy) On rough PDEs Nagoya / 23

22 Integration result Proposition Assume that x allows to build X x such that for γ > 1/2, For any α [0, 1/2) s.t. 2αp > 1, X x,i C γ (L(B α,p, B α,p )) The algebraic relation ˆδX x,i = 0 is satisfied. Consider z C 0 1(B γ,p ) C γ 1 (B p ). Then the element t s S tudx u z u 1 Is well-defined as a Young integral 2 Defines an element of C γ, linearly bounded in terms of z 3 Is limit of (t k ) Π S ttk+1 X x,i t k+1 t k z i t k along partitions Π Samy T. (Nancy) On rough PDEs Nagoya / 23

23 Perspectives Some possible or ongoing projects: 1 Study of the solutions to rough SPDEs Study of the law of yt (first result with Deya) Numerical schemes Ergodic properties 2 Evolution equations via multiparametric integration Walsh vs. Da Prato 3 Viscosity solutions: rough paths structures of sub or superjets Samy T. (Nancy) On rough PDEs Nagoya / 23