By Sergio Albeverio, Philippe Blanchard, ZhiMing Ma (auth.), Prof. Dr. U. Hornung, Prof. P. Kotelenez, Prof. G. Papanicolaou (eds.)
Read or Download Random Partial Differential Equations: Proceedings of the Conference held at the Mathematical Research Institute at Oberwolfach, Black Forest, November 19–25, 1989 PDF
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Additional info for Random Partial Differential Equations: Proceedings of the Conference held at the Mathematical Research Institute at Oberwolfach, Black Forest, November 19–25, 1989
S. Albeverio et al. 22 ProofofTh. +). 3 holds, we have for + ~ = 1. 2 yields IIPillq,oo ~ clcft- '10< t ~ 1 , for some constant Cl. 7). 11) 9n be defined in the same (Pi In, 9n) = Un, Pi9n) ~ clcft- 119nIlLf(m) IIlnllL1(m) . P Repeating the argument we obtain (for some constants Ci) Therefore we have IIPilb,2 = II(P~tlh,2 ~ cI···Cn-1 = kC~ , (t) ;:;: -ft-(1+~+ ... 14) with k := Cl ... Cn-In~. 3). 3). 2). 20) IIPillp,qp < 00 , Vt> 0 , 1< p < 00 , 1< q < 00 . 1. 2) is called LP-smoothing by Simon [Sim].
PRESUTTI (1984). L. W. Montroll, eds), 123-294.  A. OE MASI, C. KIPNIS, E. PRESUTTI, E. - Stochastics, 27, 151-165. P. - Hydrodynamieal behavior of asymmetrie particle systems. - AMS-SIAM Summer Seminar (to appear). P. - Ann. , 20, 3, 225-245.  J. GARTNER, E. (Preprint).  C. - Ann. , 13, 397 -408.  Ph. - An existenee and uniqueness result for two non strietly hyperbolie systems. (Preprint).  O. - J. Stat. , 38,1015-1025. International Series of Numerical Mathematics, Vol. A. ABSTRACT The paper is concerned with sorne approxirnate solutions for randorn partial differential equations of parabolic type via discretization in time or by an eigenfunction expansion.
5) dUt == [Aut Uo + F(ut)]dt + EtdWt , == v , where Ut==u(t,·) and v==v(·) are regarded as vectors in H==L 2 (D), Av==ßv(·), F(v)==J(·,v),Et==u(t,·) and Wt==WtO is a H-valued Brownian motion. dW. , with B( v) == Av+ F( v), the existence of a strong solution was established by Pardoux  and others. 6) successively by Euler's method based on a time-discretization. In finite dimensions, this problem was treated, for instance, in Ikeda and Watanabe . On the other hand, as PDE's (partial differential equations), the method of eigenfunction expansion or, more generally, the Galerkin method is often used to construct an approximate solution.