Singular Stochastic PDEs and Theory of Regularity Structures

Transcription

1 Page 1/19 Singular Stochastic PDEs and Theory of Regularity Structures Hao Shen (Warwick / Columbia) (Based on joint works with Martin Hairer) July 6, 2015

2 Page 2/19 Outline of talk 1. M.Hairer s theory of regularity structures: what can it do? 1.1 Well-posedness results for singular SPDEs 1.2 Universality results for dynamical models 1.3 Wong-Zakai type results for SPDE. 2. Basic ideas of the theory

3 Page 3/19 Outline of talk 1. M.Hairer s theory of regularity structures: what can it do? 1.1 Well-posedness results for singular SPDEs 1.2 Universality results for dynamical models 1.3 Wong-Zakai type results for SPDE. 2. Basic ideas of the theory

4 Page 4/19 Well-posedness of singular stochastic PDEs I Linear heat equation with space-time white t u = u + I Kardar-Parisi-Zhang (d=1) (Hairer t h = h +(@ x h) 2 + I Dynamical 4 (d=3) (Hairer t = 3 + I Sine-Gordon (d=2) (Hairer and S. t u = 1 2 u + sin( u)+ Difficulty: nonlinearities are meaningless!

5 Page 5/19 Well-posedness of singular stochastic PDEs If white noise is replaced by smooth noise ",suchthat " "!0!, then the (smooth) solutions t h " = h " +(@ x h " ) 2 + " t " = " 3 " + " ( 4 t u " = 1 2 u " + sin( u " )+ " (Sine-Gordon) do not converge to any nontrivial limit as "! 0.

6 Page 6/19 Well-posedness of singular stochastic PDEs One has to renormalize the t h " = h " +(@ x h " ) 2 A " + " (1D t " = " ( 3 " B " " )+ " (3D 4 t u " = 1 2 u " + C " sin( u " )+ " (SG 2 < 8 ) I The renormalization constants are divergent: A " " 1, B " " 1 + log ", C " " 2 /(4 ) I However, the solutions h ", ", u " converge to nontrivial limits h "! h ("! 0) etc. I These limiting solutions are the intrinsic interpretations of solutions to the singular SPDEs driven by white noise.

7 Page 7/19 Outline of talk 1. M.Hairer s theory of regularity structures: what can it do? 1.1 Well-posedness results for singular SPDEs 1.2 Universality results for dynamical models 1.3 Wong-Zakai type results for SPDE. 2. Basic ideas of the theory

8 Dynamical models for interface growth Page 8/19

9 Page 9/19 Dynamical models for interface growth Stochastic PDE model for interface growth (proposed by Kardar-Parisi-Zhang t h 2 x h {z} smoothing e ect + (@ x h) 2 {z } lateral growth + {z} space-time white noise I Kardar-Parisi-Zhang argued (physically) that this equation should describe the large scale behavior of the interface. I (Weak) Universality: Different microscopic models should all scale to this equation at large scale.

10 Page 10/19 Dynamical models for interface growth Simplest microscopic model: KPZ equation in weakly asymmetric t h 2 x h + p "(@ x h) where 1 is smooth Gaussian noise with unit correlation length. Rescale h " (x, t) =" 1 2 h(" 1 x," 2 t h " 2 x h " +(@ x h " ) 2 + " I According to well-posedness result by Hairer(2011), h " (x, t) v (") t converges to nontrivial limit as "! 0, with vertical speed v (") = " 1 C 0.

11 General class of microscopic models I (Hairer & Quastel, in t h 2 x h + p "P(@ x h)+ 1 where P is even polynomial. The rescaled solution h " has h " (x, t) v (") ver t! h h is the solution to KPZ with quadratic nonlinearity (@ x h) 2, and and v (") ver " 1 depend on all coefficients of P explicitly. I (Hairer & S. 2015) Page t h 2 x h + p "(@ x h) is smooth non-gaussian noise with unit correlation length, h " (x v hor t, t) v (") ver t! h h is the same solution to KPZ with (Gaussian) white noise; the speeds v hor, v (") ver depend on cumulants of 1 explicitly. v (") ver = " 1 C 0 + " 1 2 C1 + C 2

12 Page 12/19 Outline of talk 1. M.Hairer s theory of regularity structures: what can it do? 1.1 Well-posedness results for singular SPDEs 1.2 Universality results for dynamical models 1.3 Wong-Zakai type results for SPDE 2. Basic ideas of the theory

13 Page 13/19 Wong-Zakai theorem for SPDE I Approximating Itô solution to SDE (Wong & Zakai 1965) ẋ " = h(x " )+g(x " )Ḃ" then x " converges to Stratonovich solution. To obtain Itô solution, ẋ " = h(x " ) 1 2 g 0 (x " )g(x " ) + g(x " )Ḃ " I Approximating Itô solution to SPDE (Hairer & Pardoux t u x 2 u + H(u)+G(u) Let " be smooth Gaussian, t u " =@ x 2 u " + H(u " ) " 1 c 0 G 0 (u " )G(u " ) c 1 G 0 (u " ) 3 G(u " ) c 2 G 00 (u " )G 0 (u " )G(u " ) 2 + G(u " ) "

14 Page 14/19 Wong-Zakai theorem for SPDE Approximating Itô solution t u 2 x u + H(u)+G(u) I (Hairer & Pardoux 2014) Let " be smooth Gaussian, t u " =@ x 2 u " + H(u " ) " 1 c 0 G 0 (u " )G(u " ) c 1 G 0 (u " ) 3 G(u " ) c 2 G 00 (u " )G 0 (u " )G(u " ) 2 + G(u " ) " I Let " be smooth non-gaussian, t u " =@ x 2 u " + H(u " ) H 1 (u " ) H 2 (u " ) + G(u " ) " H 1(u ") = " 1 c 0G 0 (u ")G(u ") " 1 2 c (1) G 0 (u ") 2 G(u ") " 1 2 c (2) G 00 (u ")G(u ") 2 H 2(u) = c ( ) G 000 (u)g(u) 3 c ( ) G 0 (u) 3 G(u) c ( ) G 00 (u)g 0 (u)g(u) 2 (Work in progress by Chandra, Hairer & S.)

15 Page 15/19 Outline of talk 1. M.Hairer s theory of regularity structures: what can it do? 1.1 Well-posedness results for singular SPDEs 1.2 Universality results for dynamical models 1.3 Wong-Zakai type results for SPDE 2. Basic ideas of the theory

16 Page 16/19 Basic ideas of the theory KPZ t h = h +(@ x h) 2 + Difficulty: h 2 C 1 2 so (@ x h) 2 can t be classically defined. We illustrate the basic ideas using SDE dz t = F (Z t )db t F 2 C 1 The same difficulty: F (Z) 2 C 1 2, db 2 C 1 2, their product can t be classically defined.

17 Page 17/19 Basic ideas of the theory SDE: dz t = F (Z t )db t F 2 C 1 F (Z) 2 C 1 2, db 2 C 1 2, product can t be classically defined. I Fix t 2 R +.ThereexistsanumberY t s.t. for s t Z s Z t Y t (B s B t )+ (something smoother) F (Z s ) F (Z t ) also behaves like B s B t. I One can more or less replace F (Z)dB by BdB+(something smoother)db The only mission is to define BdB.

18 Page 18/19 Basic ideas of the theory I Differentiable function f 2 C 2 f (y) f (x) 1 + f 0 (x) (y x)+ 1 2 f 00 (x) (y x) 2 f (x) =f (x)1 + f 0 (x)x f 00 (x)x 2 I Stochastic ODE dz t = F (Z t )db t Z(s) Z(t)+Y (t)(b s B t ) Z = Z1 + Y B and for the right hand side of equation F (Z)dB = F (Z)dB + F 0 (Z)Y BdB

19 equation a little bit greater than 3/2. Therefore, we only need to keep track of its solution H Basic ideas the theory3/2. By repeatedly applying the identity (15.8), we see up to terms of of homogeneity 2 h + that solution H (@ 2 xdh)for 4C (Cis1necessarily + C2 + 4C close of the x h + to 3/2 3 ).form KPZ equation = h 1out + that + (15.7) + h Xcan 2h,in D as soon as 1 + 2be + of. By Theorem 14.5, ithturns solved 2 ttle bit for greater 3/2. Therefore, need track its h = h and hwe +honly (@ to some than real-valued functions. (Note that h keep is treated as of an independent x h) + to termsfunction of homogeneity 3/2. do Bynot repeatedly applying thehidentity (15.8), Our we here, we certainly suggest that the function is differentiable! Local behavior of analogy solutionwith to the KPZ is described by notationh is only classical Taylor immediate the solution 2 Dby for close enough to 3/2 isexpansion...) necessarilyasofanthe form is given by H = h1 + = + + h X h h + 2h,, (15.23) some real-valued and h. (Note is treated as anhand independ as an element functions of D for hsufficiently close tothat 1/2.hSimilarly, the right side of Right hand side of equation ction here, we certainly not suggest the equation is given do up to order 0 by that the function h is differentiable! O ation is only 2by analogy with the classical Taylor expansion...) As2an immedi (@H) + = h h + 4h + (h ) 1. (15.24) is given by It follows from the definition of M that one then has the = + Thank + h 1you! h 4C0,, (15. n element of D for sufficiently close to 1/2. Similarly, the right hand side so that, as an element of D with very small (but positive), one has the identity equation is given up to order 0 by Page 19/19