Ultrapotentials and positive eigenfunctions for an absolutely continuous resolvent of kernels

Abstract

Intro ductionLet (X, 0) be a measurable space and f be a submarkovian resolvent of kernels (with the initial kernel V proper) on X which is absolutely continuous and has a dual resolvent (with the same properties) with respect to a σ-finite measure.A positive numerical function s on X is called V-ultrapotentίal if it is ^-excessive (in particular ^-a.e. finite) and if the following condition is fulfilled: for every integer n >, 1, there exists a positive ^-measurable function f n on X such that s = V n (f n ), where V n is the n-th iteration of the kernel V.The main purpose of this paper (see Theorem 3.5 and Corollary 3.6) is to prove that, under a "regularity" condition (which will be discussed in the last part of Section 2) on the resolvent iΓ, for each F-ultrapotential s there exist a finite positive Borel measure σ on the open interval ]0, oo[ and a family (s x \ <λ<oa , s x being a positive Λ-eigenfunction of V (i.e. V(s x ) = λ s x and s λ is T^-a.e. finite), for any λ > 0, such that for each is o -measurable and s(x)= [s λ (x)dσ(λ).In fact, this type of representation is given for a slightly more general class of excessive functions, as the F-ultrapotentials.An uniqueness of the representation and a converse statement are also proved.These results aiv analogous, in this context, with those obtained by M. Itό and N. Suzuki in [7] (see also [6]) for the set up of diffusion semi- Recerred May 6, 1987. 125 https://www.cambridge.org/core/terms. https://doi

Bernsteins theorem for completely excessive measures

“…We give in this paper a proof for a corresponding result for measures m which are completely excessive with respect to a general semigroup (P t ) t > 0 of kernels on a nice space (cf. Theorem 2.2), extending in this way results which were obtained by Itό in [9] and by Beznea in [4]. In this result the operator L h is replaced by (Pt+h -Pt)t>o, D is replaced by the generator of the semigroup (we only consider the integrated version of ii)), and we obtain an integral representation of m as a mixture of eigenmeasures for (P t ) (which replace the exponentials of iii) above).…”

“…c) Starting only with a resolvent (U λ ) λ>0 of kernels (which does not come necessarily from a semigroup (P t ) t >o of kernels), one can characterize completely excessive measures m by properties of (mU x ) ι>Q using Stieltjes transforms instead of Laplace transforms (cf. [3] and [4]).…”

“…b) There is an obvious analogous notion of completely excessive functions. L. Beznea characterized in [4] completely excessive functions for a resolvent (U λ ) 9 supposing hypothesis L (i.e. there exists a reference measure for the resolvent) and an additional regularity assumption.…”

“…Several authors have obtained generalizations of this result to potential theoretic settings under various regularity assumptions (cf. [4], [9], [10] and references in these papers). We give in this paper a proof for a corresponding result for measures m which are completely excessive with respect to a general semigroup (P t ) t > 0 of kernels on a nice space (cf.…”

“…The proof is based on an easy extension of Bernstein's theorem for measure valued functions (cf. Proposition (1.4)), which allows to avoid Choquet's theorem which was central in [9] and [4]. The proofs are very simple; the only non-obvious result we use is the original above Bernstein's theorem and a well known result on regularization of pseudo-kernels or the fact that bimeasures on nice spaces are in fact measures which admit a disintegration.…”