Rough Burgers-like equations with multiplicative noise

Transcription

1 Rough Burgers-like equations with multiplicative noise Martin Hairer Hendrik Weber Mathematics Institute University of Warwick Bielefeld,

2 Burgers-like equation Aim: Existence/Uniqueness for du = [ xx u + g(u) x u ] dt + θ(u) dw (t). p.2

3 Burgers-like equation Aim: Existence/Uniqueness for du = [ xx u + g(u) x u ] dt + θ(u) dw (t). u(t, x) where t [, T ] time and x [, 1] one-dimensional space, p.2

4 Burgers-like equation Aim: Existence/Uniqueness for du = [ xx u + g(u) x u ] dt + θ(u) dw (t). u(t, x) where t [, T ] time and x [, 1] one-dimensional space, u(t, x) R n vector valued, p.2

5 Burgers-like equation Aim: Existence/Uniqueness for du = [ xx u + g(u) x u ] dt + θ(u) dw (t). u(t, x) where t [, T ] time and x [, 1] one-dimensional space, u(t, x) R n vector valued, g, θ : R n R n n smooth, bounded, p.2

6 Burgers-like equation Aim: Existence/Uniqueness for du = [ xx u + g(u) x u ] dt + θ(u) dw (t). u(t, x) where t [, T ] time and x [, 1] one-dimensional space, u(t, x) R n vector valued, g, θ : R n R n n smooth, bounded, dw space-time white noise, p.2

7 Burgers-like equation Aim: Existence/Uniqueness for du = [ xx u + g(u) x u ] dt + θ(u) dw (t). u(t, x) where t [, T ] time and x [, 1] one-dimensional space, u(t, x) R n vector valued, g, θ : R n R n n smooth, bounded, dw space-time white noise, periodic boundary conditions. p.2

8 Burgers-like equation Aim: Existence/Uniqueness for du = [ xx u + g(u) x u ] dt + θ(u) dw (t). u(t, x) where t [, T ] time and x [, 1] one-dimensional space, u(t, x) R n vector valued, g, θ : R n R n n smooth, bounded, dw space-time white noise, periodic boundary conditions. Case n = 1, g(u) = u Burgers equation. p.2

9 Linear case - Regularity Linear case: g = and θ = 1: Stochastic heat equation dx = xx X + dw p.3

10 Linear case - Regularity Linear case: g = and θ = 1: Stochastic heat equation dx = xx X + dw Regularity: u(x, t) is α-hölder in x and α 2 α < 1 2 but not for α 1 2. Hölder in t for any Spatial regularity is the same as regularity of Brownian motion. p.3

11 Linear case - Regularity Linear case: g = and θ = 1: Stochastic heat equation dx = xx X + dw Regularity: u(x, t) is α-hölder in x and α 2 α < 1 2 but not for α 1 2. Hölder in t for any Spatial regularity is the same as regularity of Brownian motion. Difficulty: How to interpret the nonlinear term g(u) x u? p.3

12 Classical approach Gradient case: Assume DG = g. u is a weak solution if for ϕ smooth, periodic u(t), ϕ u, ϕ t [ ] = u(s), xx ϕ + g(u) x u, ϕ ds + t ϕ θ ( u(s) ), dw s. p.4

13 Classical approach Gradient case: Assume DG = g. u is a weak solution if for ϕ smooth, periodic u(t), ϕ u, ϕ t [ ] = u(s), xx ϕ + x G(u), ϕ ds + t ϕ θ ( u(s) ), dw s. p.4

14 Classical approach Gradient case: Assume DG = g. u is a weak solution if for ϕ smooth, periodic u(t), ϕ u, ϕ t [ ] = u(s), xx ϕ G(u), x ϕ ds + t ϕ θ ( u(s) ), dw s. p.4

15 Classical approach Gradient case: Assume DG = g. u is a weak solution if for ϕ smooth, periodic u(t), ϕ u, ϕ t [ ] = u(s), xx ϕ G(u), x ϕ ds + t ϕ θ ( u(s) ), dw s. Existence and Uniqueness OK (e.g. Gyöngy 98). p.4

16 Classical approach Gradient case: Assume DG = g. u is a weak solution if for ϕ smooth, periodic u(t), ϕ u, ϕ t [ ] = u(s), xx ϕ G(u), x ϕ ds + t ϕ θ ( u(s) ), dw s. Existence and Uniqueness OK (e.g. Gyöngy 98). If n = 1 primitive G always exists, if n 2 not. p.4

17 Unstable Approximations Observation:(Hairer, Maas 1; Hairer, Voss 1) n=1 du ε = [ xx u ε + g(u ε )D ε u ε ]dt + dw D ε = approximation of derivative, e.g. D ε u(x) = 1 ε ( u(x + ε) u(x) ). p.5

18 Unstable Approximations Observation:(Hairer, Maas 1; Hairer, Voss 1) n=1 du ε = [ xx u ε + g(u ε )D ε u ε ]dt + dw D ε = approximation of derivative, e.g. D ε u(x) = 1 ε ( u(x + ε) u(x) ). u ε converges to solution of wrong equation: dũ = [ ] xx ũ + g(ũ) x ũ + c g (ũ) dt + dw Constant c depends on the approximation. p.5

19 Second look at nonlinearity g(u) x u, ϕ = 1 g ( u(x) ) ϕ(x) d x u(x) u(x) same regularity as Brownian motion! Looks like a stochastic integral! p.6

20 Second look at nonlinearity g(u) x u, ϕ = 1 g ( u(x) ) ϕ(x) d x u(x) u(x) same regularity as Brownian motion! Looks like a stochastic integral! Observations: Extra term in the unstable approximation looks like Itô -Stratonovich correction: g (u) d[u] with "quadratic variation" [u]. p.6

21 Second look at nonlinearity g(u) x u, ϕ = 1 g ( u(x) ) ϕ(x) d x u(x) u(x) same regularity as Brownian motion! Looks like a stochastic integral! Observations: Extra term in the unstable approximation looks like Itô -Stratonovich correction: g (u) d[u] with "quadratic variation" [u]. In gradient case Itô integral: 1 DG(B t ) db t = G(B 1 ) G(B ) can be defined pathwise. 1 G(B t ) dt p.6

22 Rough integrals Use a stochastic integration theory to make sense of in non-gradient case. 1 g ( u(x) ) ϕ(x) d x u(x) p.7

23 Rough integrals Use a stochastic integration theory to make sense of 1 g ( u(x) ) ϕ(x) d x u(x) in non-gradient case. Problem: Itô theory: Convergence of i g(u ( ) i)ϕ i ui+1 u i requires that u i is adapted to a filtration. In space integral no natural time direction. Solution: "Pathwhise" approach using Lyons rough path theory. p.7

24 Outline Brief account of rough path theory à la Gubinelli Concept of solution, main result Construction of solutions p.8

25 Young integration Aim: Define 1 Y s dx s for X, Y C α, α > 1 2 : I = Y ( X1 X ) I 1 = Y ( X 1 2. I n = 2 n 1 i= ) ( ) X + Y 1 X1 X ( ) Y t n Xt n i i+1 X t n i (t n i = i 2 n ) p.9

26 Young integration Aim: Define 1 Y s dx s for X, Y C α, α > 1 2 : I = Y ( X1 X ) I 1 = Y ( X 1 2. I n = Convergence 2 n 1 i= I n I n 1 = ) ( ) X + Y 1 X1 X ( ) Y t n Xt n i i+1 X t n i 2 n 1 i= i even (t n i = i 2 n ) ( Yt n Y )( t i+1 i n Xt n X ) t i+2 i+1 n 1 2 2n X α 2 nα Y α 2 nα p.9

27 Young integration Aim: Define 1 Y s dx s for X, Y C α, α > 1 2 : I = Y ( X1 X ) I 1 = Y ( X 1 2. I n = 2 n 1 i= Theorem (Young 36) ) ( ) X + Y 1 X1 X ( ) Y t n Xt n i i+1 X t n i (t n i = i 2 n ) 1 ( Yt Y ) dxt 1 2 2α 2 X α Y α. p.9

28 Rough Paths Idea: If α < 1 2 higher order approximation is necessary! A rough paths (X, X) consists of X C α and X = X s,t such that p.1

29 Rough Paths Idea: If α < 1 2 higher order approximation is necessary! A rough paths (X, X) consists of X C α and X = X s,t such that X t,t = vanishes on diagonal. p.1

30 Rough Paths Idea: If α < 1 2 higher order approximation is necessary! A rough paths (X, X) consists of X C α and X = X s,t such that X t,t = vanishes on diagonal. Regularity: X X s,t 2α = sup s t s t 2α < p.1

31 Rough Paths Idea: If α < 1 2 higher order approximation is necessary! A rough paths (X, X) consists of X C α and X = X s,t such that X t,t = vanishes on diagonal. Regularity: X X s,t 2α = sup s t s t 2α < Consistency: X s,t X s,u X u,t = ( X u X s )( Xt X u ) p.1

32 Rough Paths Idea: If α < 1 2 higher order approximation is necessary! A rough paths (X, X) consists of X C α and X = X s,t such that X t,t = vanishes on diagonal. Regularity: Consistency: X X s,t 2α = sup s t s t 2α < X s,t X s,u X u,t = ( X u X s )( Xt X u ) Think of iterated integral: X s,t = t s ( Xu X s ) dxu p.1

33 Controlled rough paths (X, X) D α = {rough paths} fixed. Y C α is controlled by X if Y t Y s = Y s ( ) Xt X s + R Y s,t p.11

34 Controlled rough paths (X, X) D α = {rough paths} fixed. Y C α is controlled by X if Y t Y s = Y s ( ) Xt X s + R Y s,t Y C α "derivative" of Y w.r.t X. p.11

35 Controlled rough paths (X, X) D α = {rough paths} fixed. Y C α is controlled by X if Y t Y s = Y s ( ) Xt X s + R Y s,t Y C α "derivative" of Y w.r.t X. R Y remainder R Y R Y s,t 2α = sup s t s t 2α < p.11

36 Controlled rough paths (X, X) D α = {rough paths} fixed. Y C α is controlled by X if Y t Y s = Y s ( ) Xt X s + R Y s,t Y C α "derivative" of Y w.r.t X. R Y remainder R Y R Y s,t 2α = sup s t s t 2α < Y is uniquely determined if X is sufficiently rough. Example: Y t = g ( X t ) for g C 2 b. p.11

37 Rough integrals 1 Aim: Define 1 Y s dx s for (X, X) D α, Y C α X : I = Y ( X1 X ) + Y X,1. I n = 2 n 1 i= Y t n i ( Xt n i+1 X t n t n i i ) + Y X t n i,t n i+1 (t n i = i 2 n ) p.12

38 Rough integrals 1 Aim: Define 1 Y s dx s for (X, X) D α, Y C α X : I = Y ( X1 X ) + Y X,1. I n = 2 n 1 i= Y t n i ( Xt n i+1 X t n t n i i ) + Y X t n i,t n i+1 (t n i = i 2 n ) Convergence I n I n 1 = 2 n 1 i= i even ( Y t n t n Y i+1 i ) Xt n + ( i+1,tn RY i+2 ti n,ti1 n Xt n X ) t i+2 i+1 n 1 2 2n ( Y α 2 nα X 2α 2 2nα + R Y 2α 2 2nα X α 2 nα) p.12

39 Rough integrals 2 Theorem (Gubinelli 5) (X, X) D α, Y C α X for 1 3 < α < 1 2. Then 1 ( ) Yt Y dxt Y X,1 1 ( 2 3α Y α X 2α + R Y 2α X α ). 2 p.13

40 Rough integrals 2 Theorem (Gubinelli 5) (X, X) D α, Y C α X for 1 3 < α < 1 2. Then 1 ( ) Yt Y dxt Y X,1 1 ( 2 3α Y α X 2α + R Y 2α X α ). 2 Also possible to construct Y dz for Y, Z C α X strategy. with the same p.13

41 Rough integrals 2 Theorem (Gubinelli 5) (X, X) D α, Y C α X for 1 3 < α < 1 2. Then 1 ( ) Yt Y dxt Y X,1 1 ( 2 3α Y α X 2α + R Y 2α X α ). 2 Also possible to construct Y dz for Y, Z C α X strategy. with the same Agenda: Construction of 1 g( u(x) ) du(x) separated into two parts: p.13

42 Rough integrals 2 Theorem (Gubinelli 5) (X, X) D α, Y C α X for 1 3 < α < 1 2. Then 1 ( ) Yt Y dxt Y X,1 1 ( 2 3α Y α X 2α + R Y 2α X α ). 2 Also possible to construct Y dz for Y, Z C α X strategy. with the same Agenda: Construction of 1 g( u(x) ) du(x) separated into two parts: Construct for every t reference rough path (X(t), X(t)) and show that u(t) is controlled by X(t). p.13

43 Rough integrals 2 Theorem (Gubinelli 5) (X, X) D α, Y C α X for 1 3 < α < 1 2. Then 1 ( ) Yt Y dxt Y X,1 1 ( 2 3α Y α X 2α + R Y 2α X α ). 2 Also possible to construct Y dz for Y, Z C α X strategy. with the same Agenda: Construction of 1 g( u(x) ) du(x) separated into two parts: Construct for every t reference rough path (X(t), X(t)) and show that u(t) is controlled by X(t). Apply Gubinelli s result to Y = g(u) and Z = u. p.13

44 Reference rough path Solution of linear stochastic heat equation X(t, x) = t S(t s) dw s (x). S(t) = heat semigroup on [, 1]. Existence results for Gaussian rough paths can be applied. p.14

45 Reference rough path Solution of linear stochastic heat equation X(t, x) = t S(t s) dw s (x). S(t) = heat semigroup on [, 1]. Existence results for Gaussian rough paths can be applied. Theorem (Friz, Victoir 5, Hairer 1) For fixed t there is a canonical definition for X(t, x, y) = y x ( X(t, z) X(t, x) ) dz X(t, z). Furthermore t X(t, ) is a.s. continuous w.r.t 2α. p.14

46 Weak Solutions du = xx u + g(u) x u + θ(u) dw (1) p.15

47 Weak Solutions du = xx u + g(u) x u + θ(u) dw (1) A weak solution to (1) is an adapted process taking values in C([, T ], C) L 1( [, T ], C α X ) such that for every smooth, periodic test function ϕ t ϕu(t) = ϕ, u + ϕ θ ( u(s) ) t, dw (s) + xx ϕ, u(s) ds + t ( 1 Non-linear term rough integral! ϕ(x)g ( u(s, x) ) d x u(s, x) ) ds. p.15

48 Mild solutions A mild solution to (1) is an adapted process u taking values in C([, T ], C) L 1( [, T ], C α X ) such that t u(x, t) = S(t)u (x) + S(t s) θ(u(s)) dw (s)(x) t ( 1 + ˆp t s (x y) g ( u(s, x) ) ) d y u(s, y) ds. ˆp t = heat kernel on [, 1]. Every weak solution is a mild solution an vice versa. p.16

49 Main result Existence/Uniqueness: Initial data u C β for 1 3 < α < β < 1 2. g C 3, θ C 2 bounded derivatives. p.17

50 Main result Existence/Uniqueness: Initial data u C β for 1 3 < α < β < 1 2. g C 3, θ C 2 bounded derivatives. Theorem For every time interval [, T ] there exists a unique weak/mild solution to (1). p.17

51 Main result Existence/Uniqueness: Initial data u C β for 1 3 < α < β < 1 2. g C 3, θ C 2 bounded derivatives. Theorem For every time interval [, T ] there exists a unique weak/mild solution to (1). If u ε is a solution to (1) with smoothened noise, then u ε converges to u in probability. p.17

52 Main result Existence/Uniqueness: Initial data u C β for 1 3 < α < β < 1 2. g C 3, θ C 2 bounded derivatives. Theorem For every time interval [, T ] there exists a unique weak/mild solution to (1). If u ε is a solution to (1) with smoothened noise, then u ε converges to u in probability. Extends the construction of Hairer 1 to the multiplicative noise case. p.17

53 Dependence on X Different choices for X e.g. Itô -Stratonovich correction X(t, x, y) = X(t, x, y) + c(y x). p.18

54 Dependence on X Different choices for X e.g. Itô -Stratonovich correction X(t, x, y) = X(t, x, y) + c(y x). Then non-linear term becomes (for n = 1) 1 ϕ(x) g ( u(x) ) d x u(x) = lim ϕ(x i )g ( u(x i ) )( u(x i+1 ) u(x i ) ) i + ϕ(x i ) g ( u(x i ) ) θ ( u(x i ) ) 2 ( ) X(x i, x i+1 ) + c(x i+1 x i ) 1 = ϕ(x)g ( u(x) ) 1 d x u(x) + c ϕ(x)g (u(x))θ ( u(x) ) 2 dx. p.18

55 Dependence on X Different choices for X e.g. Itô -Stratonovich correction X(t, x, y) = X(t, x, y) + c(y x). Then non-linear term becomes (for n = 1) 1 ϕ(x) g ( u(x) ) d x u(x) = lim ϕ(x i )g ( u(x i ) )( u(x i+1 ) u(x i ) ) i + ϕ(x i ) g ( u(x i ) ) θ ( u(x i ) ) 2 ( ) X(x i, x i+1 ) + c(x i+1 x i ) 1 = ϕ(x)g ( u(x) ) 1 d x u(x) + c ϕ(x)g (u(x))θ ( u(x) ) 2 dx. Extra term c g (u) θ(u) 2 appears. p.18

56 We have the right solution The rough path (X, X) is geometric i.e. Sym ( X(x, y) ) = 1 2 ( ) X(x, y) + X(x, y) T = 1 δx(x, y) δx(x, y). 2 This implies that in the gradient case our solution coincides with classical solution. p.19

57 We have the right solution The rough path (X, X) is geometric i.e. Sym ( X(x, y) ) = 1 2 ( ) X(x, y) + X(x, y) T = 1 δx(x, y) δx(x, y). 2 This implies that in the gradient case our solution coincides with classical solution. In the non-gradient case even different geometric rough paths give rise to different solutions, e.g. for n = 2: X(t, x, y) = X(t, x, y) + (y x) 1 and 1 g (x, y) = sin(y) cos(y). cos(x) sin(x) p.19

58 Stochastic convolution 1 θ adapted L 2 [, 1]-valued process. Stochastic convolution: Ψ θ (t) = t S(t s) θ(s) dw (s). Same space-time regularity as X(t, x) but not Gaussian! Friz-Victoir results about Gaussian rough paths can not be applied. p.2

59 Stochastic convolution 1 θ adapted L 2 [, 1]-valued process. Stochastic convolution: Ψ θ (t) = t S(t s) θ(s) dw (s). Same space-time regularity as X(t, x) but not Gaussian! Friz-Victoir results about Gaussian rough paths can not be applied. Ψ θ is controlled by X with derivative process θ ( ) Ψ θ (t, y) Ψ θ (t, x) = θ(t, x) X(t, y) X(t, x) + R θ (t, x, y). p.2

60 Stochastic convolution 2 Additional regularity for θ [ θ(t, x) θ(s, y) p θ p,α = E sup x y,s t ( t s α/2 + x y α) p + sup x,t θ(t, x) p ] 1/p. p.21

61 Stochastic convolution 2 Additional regularity for θ [ θ(t, x) θ(s, y) p θ p,α = E sup x y,s t ( t s α/2 + x y α) p + sup x,t θ(t, x) p ] 1/p. By definition of R θ : R θ (t, x, y) = t 1 (ˆpt s (z y) ˆp t s (z x)) ( ) θ(s, z) θ(t, x) W (ds, dz). p.21

62 Stochastic convolution 2 Additional regularity for θ [ θ(t, x) θ(s, y) p θ p,α = E sup x y,s t ( t s α/2 + x y α) p + sup x,t θ(t, x) p ] 1/p. By definition of R θ : R θ (t, x, y) = t 1 (ˆpt s (z y) ˆp t s (z x)) ( ) θ(s, z) θ(t, x) W (ds, dz). Then one has for p large and ϑ small: [ E R θ ] p ) C ([,τ ϑ X α ];ΩC K 2α C ( 1 + K p) θ p p,α. p.21

63 Nonlinearity/Regularisation trick Consider v = u Ψ θ. Observation: v is more regular than u. p.22

64 Nonlinearity/Regularisation trick Consider v = u Ψ θ. Observation: v is more regular than u. For fixed controlled rough path Ψ we can solve the fixed point problem v(t, x) = t in C([, T ], C 1 [, 1]). [ S(t s) g ( v(s) + Ψ(s) ) ( ) ] x v(s) ds (x) t [ 1 ] + ˆp t s (x y)g(v(s, y) + Ψ(s, y)) d y Ψ(y, s) ds p.22

65 Nonlinearity/Regularisation trick Consider v = u Ψ θ. Observation: v is more regular than u. For fixed controlled rough path Ψ we can solve the fixed point problem v(t, x) = t in C([, T ], C 1 [, 1]). [ S(t s) g ( v(s) + Ψ(s) ) ( ) ] x v(s) ds (x) t [ 1 ] + ˆp t s (x y)g(v(s, y) + Ψ(s, y)) d y Ψ(y, s) ds Fixed point v Ψ depends continuously on Ψ. p.22

66 Outer fixed point For T small enough mapping is a contraction w.r.t. [ u p,α = E sup x y,s t u θ(u) Ψ θ(u) + v Ψθ(u) u(t, x) u(s, y) p ( t s α/2 + x y α) p + sup x,t u(t, x) p ] 1/p. p.23

67 Outer fixed point For T small enough mapping is a contraction w.r.t. [ u p,α = E sup x y,s t u θ(u) Ψ θ(u) + v Ψθ(u) u(t, x) u(s, y) p ( t s α/2 + x y α) p + sup x,t u(t, x) p ] 1/p. Local existence! p.23

68 Outer fixed point For T small enough mapping is a contraction w.r.t. [ u p,α = E sup x y,s t u θ(u) Ψ θ(u) + v Ψθ(u) u(t, x) u(s, y) p ( t s α/2 + x y α) p + sup x,t u(t, x) p ] 1/p. Local existence! Data is bounded Gronwall argument gives global existence! p.23

69 Outer fixed point For T small enough mapping is a contraction w.r.t. [ u p,α = E sup x y,s t u θ(u) Ψ θ(u) + v Ψθ(u) u(t, x) u(s, y) p ( t s α/2 + x y α) p + sup x,t u(t, x) p ] 1/p. Local existence! Data is bounded Gronwall argument gives global existence! Construction continuous in reference rough path (X, X) + Stability of Friz-Victoir construction Stability! p.23

70 Conclusion Spatial rough integrals give a way to define solutions to equations that are not well-posed classically. p.24

71 Conclusion Spatial rough integrals give a way to define solutions to equations that are not well-posed classically. Our construction gives an interpretation for extra terms that appear even in well-posed equations. p.24

72 Conclusion Spatial rough integrals give a way to define solutions to equations that are not well-posed classically. Our construction gives an interpretation for extra terms that appear even in well-posed equations. It should also give a different method to prove such approximation results. p.24

73 Conclusion Spatial rough integrals give a way to define solutions to equations that are not well-posed classically. Our construction gives an interpretation for extra terms that appear even in well-posed equations. It should also give a different method to prove such approximation results. Rough path machinery very convenient, as it allows to separate the construction into a stochastic part and a deterministic part. p.24