By C. Menini, F. van Oystaeyen

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UNIQUENESS PROBLEMS IN VARIOUS CONTEXTS for all f E Dom (L). We then conclude that the resolvent is sub-Markov, and finally obtain the sub-Markov property for the semigroup. 18). Fix f E A, and let ~b,~ : R ~ R, n E N, be smooth increasing functions such that Cn(s) = 0 for s < 1, Cn(s) = (s - 1) p-1 for s _> 1 + ~, and 0 _< @,~(s) _< ( s - 1) p-1 for all s E R. Let ~ n := f~ r ds. Then 0 _< %~(t) < ( ( t - 1)+) p / p for all t E R and n E N. , < ~ - P (f-1);dm. 19) 1) p - I L f d m < (f- /_>1} -_ (~ j[{ P ( f -- 1) p din.

REMARK. , k F (r , v) = 0r ~ ~-x/ ( U l , . . , u k ) i=1 for all k E N, r e C ~ ( R k) such that r r ( u i , V) = 0, and v, u l , . . , uk C A. From now on, we fix an abstract diffusion operator (L, A) on LP(E;rn), and c~ _> 0. , f and L f are in L I ( E ; rn), and Lfdm Lemma < a/fdm. 8 Suppose (A 1) holds. Then ( L - ~, C~ A) is dissipative on LP(E ; rn) PROOF. e. for all f c A. Hence, by the sub-invariance, f r < fL(Oof) dm <_ ~ / ~of dm ~ <_ p Ifl p dm. for all f C A. This implies the dissipativity, cf.

The forms ($Y, 5r~), y C R 1, are the Dirichlet forms of Brownian motion with reflection at y. All these forms extend (C, C ~ ( R 1 ) ) . Nevertheless, Problem 8 has a positive answer for this example. g. from the essential self-adjointness of the corresponding diffusion operator E f = fll with domain C ~ ( R 1 ) , cf. 5 below and the diagram in Section e), 2), below. In fact, the domain of the generator of (gY, jry) contains only those functions f in C ~ ( R 1) that satisfy the Neumann condition f'(y) = O.