Markov processes with identical excessive measures

Getoor, R. K.; Glover, Joseph

doi:10.1007/bf01163505

Cited by 31 publications

(20 citation statements)

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“…( (3) The resolvent of a right process satisfies (IE); furthermore it satisfies (NSP) and (UC) according to the results of [12] mentioned in the introduction. Therefore by (2), since (NSP) trivially implies (SSP), in particular the extremality of point charges (EPC) holds for right processes.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…For a general reference concerning resolvents, excessive functions and measures we refer to [4]; in connection with processes we refer as well to [2,11,7,12,10], and concerning especially Ray resolvents and Ray compactifications to [11,7,21] in addition to [4].…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…Given a right process with semigroup (PJ in the sense of [11], the following three facts about the set of excessive measures are well known: Proofs of (0.1) and (0.2) can be found in the paper [12] by Getoor and Glover. As to result (0.3), mentioned already by Dynkin [6], several different proofs have been published recently, by Fitzsimmons and Maisonneuve [10], Getoor and Glover [14], Janssen [16], and again Fitzsimmons [9].…”

Section: O Introductionmentioning

confidence: 99%

“…It seems that in all approaches so far the set of excessive measures has not been considered; the reason may be that interest in properties of excessive measures as opposed to excessive functions has been renewed only recently (see, e.g., [6,12,13,10]). …”

Section: O Introductionmentioning

confidence: 99%

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Excessive Measures and the Existence of Right Semigroups and Processes

Steffens¹

1989

Transactions of the American Mathematical Society

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ABSTRACT. Given a resolvent (Un) on a Lusin space (E. IE) , the paper gives necessary and sufficient conditions in terms of the excessive measures that ensure the existence of a right process, resp. a right continuous semigroup, on (E. IE) with resolvent (Un). Furthermore, a notion of nonbranch points with respect to (U,,) is introduced-also in terms of the excessive measures-and various characterizations are given. They show, in particular, the equivalence of this definition with those introduced and discussed by Engelbert and Wittmann.

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Section: Notation and Preliminariesmentioning

confidence: 99%

Section: Notation and Preliminariesmentioning

confidence: 99%

Section: O Introductionmentioning

confidence: 99%

Section: O Introductionmentioning

confidence: 99%

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Excessive Measures and the Existence of Right Semigroups and Processes

Steffens¹

1989

Transactions of the American Mathematical Society

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“…But even In this 6 1. 6 10 Normalize 0 so that sup({(x): xEH) = I. Let W be the inverse of -A on H -so W < V. Asstne VO is bounded by a constant m, and set Fk = co for some constant c to be determined below.…”

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confidence: 99%

Positive solutions of systems of semilinear elliptic equations: the pendulum method

Glover¹

1987

Trans. Amer. Math. Soc.

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Conditions are formulated which guarantee the existence of positive solutions for systems of the form \[ − Δ u 1 + f 1 ( u 1 , … , u n ) = μ 1 , − Δ u 2 + f 2 ( u 1 , … , u n ) = μ 2 , ⋮ ⋮ ⋮ ⋮ − Δ u n + f n ( u 1 , … , u n ) = μ n , \begin {gathered} - \Delta {u_1} + {f_1}({u_1}, \ldots ,\,{u_n}) = {\mu _1}, \hfill \\ - \Delta {u_2} + {f_2}({u_1}, \ldots ,\,{u_n}) = {\mu _2}, \hfill \\ \vdots \quad \quad \quad \quad \quad \vdots \quad \quad \quad \vdots \quad \vdots \hfill \\ - \Delta {u_n} + {f_n}({u_1}, \ldots ,\,{u_n}) = {\mu _n}, \hfill \\ \end {gathered} \] , where Δ \Delta is the Laplacian (with Dirichlet boundary conditions) on an open domain in R d {\mathbf {R}^d} , and where each μ i {\mu _i} is a positive measure. The main tools used are probabilistic potential theory, Markov processes, and an iterative scheme which is not a generalization of the one used for quasimonotone systems. Quasimonotonicity is not assumed and new results are obtained even for the case where ∂ f k / ∂ x j > 0 \partial {f_k}/\partial {x_j} > 0 for every k k and j j .

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