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6) it follows that T l~m o~= ~4 f e-~' Pv (0 A (V(0)-- 2V(0 A~ (0--2~ (0 A (V(0) 0 + 2o (u (0, ~ ( t ) ) ~ - c I v (0 I~ - 2 (B~ (0, B (v (0))Q + t B (~ (t)) 1~] at. B=(t)--B(g(t))l~} dt-[-;e-Cr40. 17), we see that B~ (t) = B(v(t)) [ a . e . (t, w)], and 8:Iim M[av(T)[~--MI~(T)I~=O. < O. 11), a n d l e t y = v - Xx, ? , ~ R + . < O. < 0. 0 Since x i s a r b i t r a r y , f r o m t h i s it f o l l o w s t h a t A~ (t) = A (v(t)) [a. e. 7) and the f a c t p r o v e d e a r l i e r t h a t Boo (t) B(v(t)) [a.
1. 2) is s a t i s f i e d , and f o r any ~ E V t (v (t), rl)o~(Uo, ~])0--S (A~,. ~ (D~,. . D ~ v , s), D ~ . . D ~ l ) o d s 0 t -t- ~ ~ B (Dh. . : (t, ~)). 3) in the sense of the i n t e g r a l identity. 1) and the fact that f o r , e H, the n o r m 113 (13p l . . D p m v ( t , x), t, x)IE m u l t i p l i e d by I~/(x)l is integrable o v e r R d, and hence f o r any e ~ E ~B (v, t) e = (~, B (v, t) e). D~",v, t, x), e)e~ (x) d x Rd ~ (~a B (D,, . . Dgrn v, t, x) ~ (x) d x , e)E. 3). 2.
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Stochastic evolution equations by Krylov