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Let (d ) -_. 5. (xfO =* £ for x € D , and that H^f € C,(D) if 6 > 0 is compact and that f € C,(D) . Then H f Dirichlet problem w? th boundary data fining H_ f = f on U compact, open subsets of . (If U D , then in the sense of Hansen ). is nothing but the solution of the fi . , which is extended to runs through a basis (H ) -.. o U D by de- of relatively is a family of harmonic kernels Now we want to formulate and solve a Dirichlet problem for ("sufficiently many") distributions. A natural way to do this is to approximate a given distribution Cn £ € V% , which represents the "boundary data", by smooth functions , n € IN , and tryJ to get & a limit for (HTT -__ U£n )n€IN in V1 .

H d y ( x ) < 2" E Proof. Let n €U be such that K U supp y c U n k . , x - || w ° * ||E , x e u , sucn tnat DIRICHLET PROBLEM AND RANDOM FIELDS are y - i n t e g r a b l e . By 2 . 9 ( i i ) we have f o r 29 n > nQ K U n -»*"E -<2||^o|| E . 9 (iii) and the theorem of dominated convergence imply the assertion. a We see that the subsequence hence on K. 3 depends on is fixed we set from now on for simplicity (U ) ^_, := (U_ ). Cm . 6). Fix y , (U ) ^ Q (U,K) as before in I , it will satisfy the properties P € Bi .

I) Let n € IN and ip 6 P(K U U) . Then n U supp(

- - X 0c for every Proof, (ii): (£) n £ € P* . 2 (i). We may assume that cp > 0 . Let £ € P 1 . Then = | H u (£ n )(x) (U) 6 a(U C ) and P(Q and cp € P, . 3 this implies V.

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