Wiener Chaos Solution of Stochastic Evolution Equations

Transcription

1 Wiener Chaos Solution of Stochastic Evolution Equations Sergey V. Lototsky Department of Mathematics University of Southern California August 2003 Joint work with Boris Rozovskii

2 The Wiener Chaos (Ω, F, {F t } t 0, P), W (t) = (w k (t), k 1). {m i (s), i 1} CONS in L 2 ([0, T ]), ξ ik = J = T ξ α = i,k m i(s)dw k (s). { 0 α = (α k i, i, k 1) α = i,k H α k(ξ ik ) i α k i!, where α k i < }; H n (x) = ( 1) n e x2 /2 dn dx ne x2 /2. Note: if α = 0, then ξ α = 1. 1

3 Theorem. (Cameron & Martin, 1947) The collection {ξ α, α basis in L 2 (Ω, FT W, P): J} is an orthonormal If η L 2 (Ω, F W T, P) and η α = E(ηξ α ), then η = α J η α ξ α and E η 2 = α J η 2 α. 2

4 A Technical Lemma Lemma. Define ξ α (t) = E(ξ α Ft W ). Then dξ α (t) = α k i ξ α (i,k) (t)m i(t)dw k (t), i,k where α (i, k) is the multi-index with the components ( ) { l α max(α k (i, k) = i 1, 0), if i = j and k = l, j α l j, otherwise. Note: If α = 0, then ξ α (t) = 1 for all t 0. Otherwise, ξ α (0) = 0. 3

5 Example 1: SPDE With a Classical Solution du(t, x) = 1 u xx (t, x)dt + 1 u x (t, x)dw(t), t > 0, x R, u(0, x) = ϕ(x) L 2 (R) = 0.5 > 0. Fact: sup 0<t<T E u 2 L 2 (R) (t) T 0 E u x 2 L 2 (R) (s)ds ϕ 2 L 2 (R). Then u(t, x) = α J u α (t, x)ξ α. 4

6 Example 1: The S-system One Wiener process, so α = (α i, i 1). dξ α (t) = αi ξ α (i) (t)m i(t)dw(t), i ξ α (0) = I( α = 0); du(t, x) = u xx (t, x)dt + u x (t, x)dw(t). By Ft W -measurability of u, u α (t, x) = E(u(t, x)ξ α (t)); u α = (u α ) xx + i αi (u α (i) ) xm i (t), Also, sup 0<t<T u α (0, x) = ϕ(x)i( α = 0) α J α J u α 2 L 2 (R) (t) T 0 (u α) x 2 L 2 (R) (s)ds ϕ 2 L 2 (R). 5

7 The WC Method Equation S-System Solution of the S-System The Winer Chaos Solution of the Equation as a (Formal) Fourier Series. Theorem. If the Wiener Chaos solution belongs to the traditional solution space, then it is a traditional solution. For du = u xx dt + u x dw(t), traditional solution means (u, ψ)(t) = (ϕ, ψ) t 0 (u x, ψ)(s)dw(s), ψ C 0 (R). t 0 (u x, ψ x )(s)ds + 6

8 Example 2: SPDE Without Classical Solution du(t, x) = u xx (t, x)dt + 2u x (t, x)dw(t), t > 0, x R, u(0, x) = ϕ(x). No classical solution: (2 2 ) = 1 < 0. Wiener Chaos solution: u(t, x) = u α (t, x)ξ α, α u α = (u α ) xx + 2 i αi (u α (i) ) xm i (t), u α (0, x) = ϕ(x)i( α = 0). Now u α 2 L 2 (R)(t) =, t > 0, but α J sup 0<t<T 4 k k 0 α =k u α 2 L 2 (R) (t) < ϕ 2 L 2 (R). 7

9 The S-System Hilbert spaces H s, H f, H i. Operators A, M k : H s H f. Assume v(t) = Av(t) + f(t) has a unique solution v L 2 ((0, T ); H s ) for every f L 2 ((0, T ); H f ) and v(0) H i. u α (t) = Au α (t) + i,k α k i M ku α (i,k) (t)m i(t), u(t) = α u α (0) = u 0 I( α = 0) H i. u α (t)ξ α Wiener Chaos solution of du(t) = Au(t)dt + k 1 M k u(t)dw k (t), u(0) = u 0. 8

10 Solving the S-System Theorem. Assume that the operator A generates a semi-group (T t, t 0) so that, for t > 0, the operators T t and M k T t are bounded in a Hilbert space H. Let u 0 H. Then, for every N 0 and 0 < t < T, u α 2 H (t) α =N = t k 1,...,k N sn s2 0 T t sn M kn... T s2 s 1 M k1 T s1 u 0 2 H ds 1... ds N. 9

11 Example 3 du(t, x) = u xx (t, x)dt + b(x)u x (t, x)dw(t), t > 0, x R, u(0, x) = ϕ(x) L 2 (R), b = b(x) measurable, sup b(x) 2 2. Then M = b(x) / x and α N u α 2 L 2 (R) (t) ϕ 2 L 2 (R) t 0 s 0 sn 0 x R... s2 0 MT t sn M... T s2 s 1 MT s1 ϕ 2 L 2 (R) ds 1... ds N ds. In particular, α u α 2 L 2 (R) (t) ϕ 2 L 2 (R) and u(t) L 2 (Ω; L 2 (R)) for all t [0, T ]. 10

12 Weighted Wiener Chaos Idea: replace w k with q k w k. Q = {q 1, q 2,...}, q k > 0; Q Q means q k q k for all k; q α = q αk i k. i,k Definition. For a Banach space X, the Q- weighted Wiener Chaos space L 2,Q (FT W ; X) is L 2,Q (F W T ; X) = (u α) : α J q 2α u α 2 X <. Q = 1 L 2,Q (F W T ; X) = L 2(F W T ; X); Q Q L 2, Q (FW T ; X) L 2,Q(F W T ; X). 11

13 Example 4: w k q k w k 1. (Obvious) If u(t) = 1 + u(s)dw k(s), k 1 0 then u L 2,Q (FT W ; R) for every Q = (q 1, q 2,...) so that qk 2 <. k 1 t 2. (Nualart-Rozovskii, 1997) If du(t, x) = u(t, x)dt + u(t, x)dw(t, x), t > 0, x R d, d 2, then u L 2,Q (F W T ; L 2(R d )) for some Q. 12

14 Theorem. Consider du(t, x) = u xx (t, x)dt + b(x)u x (t, x)dw(t), t > 0, x R, u(0, x) = ϕ(x) L 2 (R), b = b(x) bounded, measurable; Q = (q, 1, 1,...), q > 0. Then u(t) L 2,Q (FT W ; L 2(R)) q 2 sup b(x) 2 < 2. x R If sup b(x) 2 < 2, then u(t) L 2,Q (FT W ; L 2(R)) x R for some q > 1. If sup b(x) 2 2, then u(t) L 2,Q (FT W ; L 2(R)) x R for some q < 1. 13

15 S-Transform h(t) = (h 1 (t),... h n (t)); E(h) = exp T k 0 h k(t)dw k (t) 1 2 k h k 2 L 2 ((0,T )) If h k (t) = n k h αk i ik α k i h ik m i (t), then (E(h)) α = i=1 i,k E(h) 2 L 2,Q (FT W ;R) = exp { i,k h 2 } i,k q2 k < for every Q. For u L 2,Q (FT W ; X) can then define u h = α u α (E(h)) α S-transform of u. ; 14

16 Definition. Soft solution of du(t) = Au(t)dt + k 1 M k u(t)dw k (t) means u h (t) = Au h (t) + k 1 h k (t)m k u h (t). Theorem. If u L 2,Q (FT W ; X) for some Q, then the soft solution is equivalent to the Wiener Chaos solution. To study the properties of the solution, it is easier to work with (u α ) than with u h. Can have Wiener Chaos solution that belongs to no L 2,Q. 15

17 Example 5 du(t, x) = u x (t, x)dw(t), t > 0, x R, u(0, x) = ϕ(x) S(R). Then α =N u α 2 tn L 2 (R)(t) = N! ϕ(n) x 2 L 2 (R). ϕ (N) x 2 L 2 (R) CN N! u(t) L 2 (Ω; L 2 (R)). ϕ (N) x 2 L 2 (R) CN N! u(t) L 2,Q(t) (Ω; L 2 (R)). ϕ (N) x 2 L 2 (R) ean 2 u(t) / L 2,Q (Ω; L 2 (R)). 16

18 References Wiener Chaos: Wiener (1930 s), Cameron & Martin (1947), Hida et al. (1993), Øksendal et al. (1996) S-transform: Kondratiev et al. (1996) SODE: Krylov & Veretennikov (1976) Filtering: Wong (1981), Ocone (1983), Mikulevicius & Rozovskii (1993), Budhiraja & Kallianpur (1996) Flows: LeJan and Raimond (2002) Soft Solutions: Mikulevicius & Rozovskii (1994), Nualart & Rozovskii (1997), Potthoff et al. (1998) Navier-Stokes equation: Mikulevicius & Rozovskii (2001) 17

19 So What? Have a pretty general procedure for constructing solutions for linear equation. S-system is often more convenient than the underlying equation. Big plans for the NS equation. 18