Finding Limits of Multiscale SPDEs

Transcription

1 Finding Limits of Multiscale SPDEs David Kelly Martin Hairer Mathematics Institute University of Warwick Coventry UK CV4 7AL November 3, 213 Multiscale Systems, Warwick David Kelly (Warwick) Multiscale SPDEs November 3, / 1

2 A Classic Result Let X ε satisfy the SDE dx ε (t) = 1 ε b(x ε/ε)dt + σ(x ε /ε)db(t) with b, σ being periodic C 2 functions and X ε taking values on S 1. Eg. Take σ = 1, b = V ( ). Then the SDE described a gradient flow over the highly oscillatory potential V ( /ε). David Kelly (Warwick) Multiscale SPDEs November 3, / 1

3 A Classic Result Let X ε satisfy the SDE dx ε (t) = 1 ε b(x ε/ε)dt + σ(x ε /ε)db(t) with b, σ being periodic C 2 functions and X ε taking values on S 1. Eg. Take σ = 1, b = V ( ). Then the SDE described a gradient flow over the highly oscillatory potential V ( /ε). If σ is strictly positive and b/σ 2 dx = then X ε c X where X is a BM and c > is some constant determined by b and σ. David Kelly (Warwick) Multiscale SPDEs November 3, / 1

4 Homogenization of PDEs If we denote the generator of X ε as L ε = 1 ε b(x/ε) x σ2 (x/ε) x 2 where x S 1. We can use the classic result to homogenize PDEs t u ε = L ε u ε t u = c x 2 u u ε () = g u() = g since u ε (t) = E[g(X ε (t))]. David Kelly (Warwick) Multiscale SPDEs November 3, / 1

5 Homogenization of PDEs If we denote the generator of X ε as L ε = 1 ε b(x/ε) x σ2 (x/ε) 2 x where x S 1. We can use the classic result to homogenize PDEs t u ε = L ε u ε t u = c 2 x u u ε () = g u() = g since u ε (t) = E[g(X ε (t))]. We can also add a forcing term t u ε = L ε u ε + f t u = c 2 x u + f u ε () = g u() = g since u ε (t) = E[g(X ε (t))] + E[f (X ε(t s), s)]ds. This works provided f = f (x, t) is nice enough. David Kelly (Warwick) Multiscale SPDEs November 3, / 1

6 Homogenization of SPDEs We try to find a limit to du ε (t) = L ε u ε (t)dt + Q 1/2 dw (t) u ε () = where dw dt is space-time white noise and Q 1/2 is a positive, bounded linear operator with Q 1/2 e ikx = λ k e ikx with λ k. Hence, we can also write du ε (t) = L ε u ε (t)dt + λ k e ikx dw k (t) k Z where W k are complex BMs with W k = W k and otherwise independent. David Kelly (Warwick) Multiscale SPDEs November 3, / 1

7 First Result Theorem (Compact Q) If λ k and u satisfies then for any s > 3/4. du(t) = c 2 x u(t)dt + Q 1/2 dw (t), E sup u ε (t) u(t) 2 H s as ε, t [,T ] If we know λ k k α for some α > then we can improve to s > (3/4 3α/2). David Kelly (Warwick) Multiscale SPDEs November 3, / 1

8 Second Result Theorem (Bounded Q) If λ k λ and û satisfies dû(t) = c 2 x û(t)dt + k ( λk 2 + λ 2 ( ρ 2 1) ) 1/2 e ikx dŵ k (t) for some new sequence of BMs Ŵ k and ρ satisfies L ρ = with ρ, 1 = 1. dist Then there exists a sequence of processes û ε = u ε such that for any s > 1. E sup û ε (t) û(t) 2 H s as ε, t [,T ] eg. For STWN, dû = c 2 x ûdt + ρ dŵ David Kelly (Warwick) Multiscale SPDEs November 3, / 1

9 Sketch of Proof We use the following interpolation strategy E u ε (t) u(t) 2 H s = for some β (, 1). For the high modes m <ε β E u ε (t) u(t), e imx 2 (1 + m 2 ) s + m ε β E u ε (t) u(t), e imx 2 (1 + m 2 ) s m ε β E u ε (t) u(t), e imx 2 (1+m 2 ) s ε 2βs ( E uε (t) 2 + E u(t) 2). If E u ε (t) 2 doesn t blow up too much, then we need only worry about the low modes. David Kelly (Warwick) Multiscale SPDEs November 3, / 1

10 Sketch of Proof We have a mild solution given by u ε (t) = k λ k S ε (t s)e ikx dw k (s) where S ε is the semigroup generated by L ε. David Kelly (Warwick) Multiscale SPDEs November 3, / 1

11 Sketch of Proof We have a mild solution given by u ε (t) = k λ k S ε (t s)e ikx dw k (s) where S ε is the semigroup generated by L ε. We take m < ε β and try to approximate u ε, e imx = k λ k S ε (t s)e ikx, e imx dw k (s) This can be approximated using (a quantitative version of) our classical result S ε (t s)e ikx = e ikx k2 (t s) + O(εk) David Kelly (Warwick) Multiscale SPDEs November 3, / 1

12 We have that u ε (t), e imx = λ m e m2 (t s) dw m (s) + λ k Rε k (t s), e imx dw k k = u, e imx + λ k Rε k (t s), e imx dw k (s) k Estimates for Rε k get bad when k ε 1, so this only works if λ k k 1/2. David Kelly (Warwick) Multiscale SPDEs November 3, / 1

13 We have that u ε (t), e imx = λ m e m2 (t s) dw m (s) + λ k Rε k (t s), e imx dw k k = u, e imx + λ k Rε k (t s), e imx dw k (s) k Estimates for R k ε get bad when k ε 1, so this only works if λ k k 1/2. A better approach is to use the adjoint semigroup u ε (t), e imx = k λ k e ikx, Sε (t s)e imx dw k (s) With the adjoint classical result S ε (t s)e imx = ρ(x/ε)e imx m2 (t s) + O(εm) David Kelly (Warwick) Multiscale SPDEs November 3, / 1

14 We have u ε (t), e imx = k λ k e ikx, ρ(x/ε)e imx e m2 (t s) dw k (s) + R Since ε 1 N, only terms with k = m + p/ε for any p Z will remain. David Kelly (Warwick) Multiscale SPDEs November 3, / 1

15 We have u ε (t), e imx = k λ k e ikx, ρ(x/ε)e imx e m2 (t s) dw k (s) + R Since ε 1 N, only terms with k = m + p/ε for any p Z will remain. We have λ m+p/ε ρ, e ipx e m2 (t s) dw m+p/ε (s) p and isolating the first term λ m e m2 (t s) dw m (s) + λ m+p/ε ρ, e ipx e m2 (t s) dw m+p/ε (s) p If λ k, this proves the result for Q compact. David Kelly (Warwick) Multiscale SPDEs November 3, / 1

16 If λ k λ, then these extra terms no longer vanish. We have that λ m+p/ε ρ, e ipx e m2 (t s) dw m+p/ε (s) is equal in distribution to p ( ) 1/2 λ m+p/ε 2 ρ, e ipx 2 e m2 (t s) dŵ m (s) p ( λ m 2 + λ 2 ( ρ 2 1) ) 1/2 This proves the result for Q bounded. e m2 (t s) dŵm(s) David Kelly (Warwick) Multiscale SPDEs November 3, / 1

17 Ergodic Properties If µ ε and µ are (respectively) the invariant measures of the original and limiting SPDEs then we can show that under the H s Wasserstein metric. µ ε µ as ε David Kelly (Warwick) Multiscale SPDEs November 3, / 1

18 Extensions Similar results hold for all SPDEs of the form du ε (t) = L ε u ε (t)dt + k q k (x/ε)e ikx dw k (t) for C 1 periodic functions q k. This is the structure possessed by noise that is cell-translation invariant. i.e. The law of the noise is invariant if you shift it by a cell. David Kelly (Warwick) Multiscale SPDEs November 3, / 1

19 Extensions Similar results hold for all SPDEs of the form du ε (t) = L ε u ε (t)dt + k q k (x/ε)e ikx dw k (t) for C 1 periodic functions q k. This is the structure possessed by noise that is cell-translation invariant. i.e. The law of the noise is invariant if you shift it by a cell. Thank You! David Kelly (Warwick) Multiscale SPDEs November 3, / 1