By Michael Rockner

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S. on V% V 1*1 k 1=1 a. i»l l X A y£n i vc E are Gaussian distributed, it o M „ (A) . M,P V i * V 3. 1. M,Pyv n' * (X An) c™ converges to X A Mi n \ii 1 < i < k . 12 ] that the sequence

L (U,u),ll II ) . Now y#(£i ) uniformly on K. 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.

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 . ) Q(U) 6 A(U C ) Proof. 12. Proposition. Pf I= and Hy m ^^'^ is * C€ P. A(UC)/8-measurable. We intend to show that G(U) = {C€«,(U) nn 0 (U) : |