A Dirichlet Problem for Distributions and Specifications for by Michael Rockner

By Michael Rockner

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5 (vi) we know that £ € ft AV ) for every lim - <£,cp> n-*» * n for cp € Pj . Hence £ € « 2 (U) and lL(£) = £ . n € IN and that . 52 MICHAEL ROCKNER (vii) is an obvious consequence of (ii) and (vi). 11. Definition. Let a £ € Pf . We say that the Dirichlet problem is solv- able for £ if £ € ft(U) , and call H (£) problem for U with "boundary data" £ . the solution of the Dirichlet Now we come to the question of how big ft(U) really is. H^ : fi(U) -> P' We extend the map extension again by H to the whole of .

5. 13. Theorem. (i) Let £ € V\ Then x-u^(£), is harmonic on (ii) If U. (U) i bourhood of 8U then and n € V1 is such that n € ft . (U) and £ € V1 (U) cone. classical PWB-solution. (iii) If x € U, then n - £ y (£) = v (n) £ € ft (U) and C)I on some open neighfor every x -> u U (£) , x € U, x x € U . is the 26 MICHAEL R&CKNER (iv) If U neighbourhood then (v) V of C € n c j (U) For x € U U and such that X \j the function A Xy c L M Let is such that there exists an open £| v is represented by a harmonic function y ^ U ) - C|v(x) o(3U)-measurable version of Remark.

Also [13]): The formulation of the classical prediction problem for an ordinary (time indexed) process is based on the notion of the "past" and the "future" of the process (cf. g. [14, Chap. 4 ] , replace the "past" for a given o(U ) and the "future" by In our situation it is natural to U c D, U open and relatively compact, by a(U). Then the prediction problem is the follow- ing: What is the conditional distribution of "past" TT a(U )? Ep D (x|a(U C )) = TTyX (P',B) supp (p c U, given the Th is can be solved by finding a probability kernel : V1 x B -*B.

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