R. K. Getoor

Markov processes : Ray processes and right processes

Preliminaries.- Resolvents.- Ray resolvents and semigroups.- Increasing sequences of supermartingales.- Processes.- Processes continued.- Characterization of previsible stopping times.- Some topology and measure theory.- Right processes.- The ray knight compactification.- Comparison of processes.- Right processes continued: Shihs theorem.- Comparison of (Xt?) and (X t? * ).- U-spaces.- The ray space.

Riesz decompositions in Markov process theory

Riesz decompositions of excessive measures and excessive functions are obtained by probabilistic methods without regularity assumptions. The decomposition of excessive measures is given for Borel right processes. The results for excessive functions are formulated within the framework of weak duality. These results extend and generalize the pioneering work of Hunt in this area.

Smooth measures and continuous additive functionals of right Markov processes

The Revuz correspondence sets up a bijection between the class of positive continuous additive functionals of a Markov process and a certain class of “smooth” measures on the state space of the process. We consider the correspondence in the context of a Borel right process with a distinguished excessive measure. A “nest” type characterization of smooth measures is provided, as well as a capacitary characterization of nests. Our results extend work of Revuz, Fukushima, and others.

Occupation time distributions for Lévy bridges and excursions

Let X be a one-dimensional Levy process. It is shown that under the bridge law for X starting from 0 and ending at 0 at time t, the amount of time X spends positive has a uniform distribution on [0, t]. When 0 is a regular point, this uniform distribution result leads to an explicit expression for the Laplace transform of the joint distribution of the pair (R, AR), where R is the length of an excursion of X from 0, and AR is the total time X spends positive during the excursion. More concrete expressions are obtained for stable processes by specialization. In particular, a formula determining the distribution of AR/R is given in the stable case.

Measure Perturbations of Markovian Semigroups

The perturbation of the semigroup of a Borel right process by a class of signed measures on the state space of the process is studied. The perturbation is defined by a Feynman–Kac functional associated with the measure. Under appropriate conditions the perturbed semigroup is strongly continuous in Lp(m), 1 ≤p<∞ where m is a fixed excessive measure. Both existence and uniqueness of the associated Schrödinger type equation are investigated.

Measures not charging semipolars and equations of schrödinger type

SupposeX is a Borel right process andm is a σ-finite excessive measure forX. Given a positive measure μ not chargingm-semipolars we associate an exact multiplicative functionalM(μ). No finiteness assumptions are made on μ. Given two such measures μ and ν,M(μ)=M(ν) if and only if μ and ν agree on all finely open measurable sets. The equation (q−L)u+uμ=f whereL is the generator of (a subprocess of)X may be solved for appropriatef by means of the Feynman-Kac formula based onM(μ). Both uniqueness and existence are considered.

Seminar on stochastic processes, 1987

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The Blumenthal-Getoor-McKean Theorem Revisited

The Blumenthal-Getoor-McKean theorem [BGM] (hereafter referred to as BGM) states that if X and \( \tilde{X} \) are two Markov processes with the same hitting distributions, then they may be time changed into each other. This is a deliberately loose statement and one needs to specify the precise hypotheses on X and \( \tilde{X} \) and also exactly what the conclusion means before it makes mathematical sense. In §V-5 of [BG] a precise statement and proof are given when X and \( \tilde{X} \) are standard processes as defined in [BG]. It is stated in several places in the literature that the proof in [BG] carries over to the case in which X and \( \tilde{X} \) are right processes. However, a careful reading of that proof reveals that the quasi-left-continuity (qlc) of X and \( \tilde{X} \) is used in a crucial manner at two points: the proofs of (V-5.4) and (V-5.20) in [BG]. The purpose of this paper is to give a careful proof of BGM for arbitrary right processes X and \( \tilde{X} \) as defined in [S].

Capacity Theory Without Duality

SummaryThe paper develops a theory of capacity for a Borel right process without duality assumptions. The basic tool in this approach is a stationary process ralative to an excessive measure.IfPt)t≧0 denotes the semigroup of the process on the state spaceE and ifm is an excessive measure onE, then there exists a processY = (Yt)t∈ℝ onE with random birth and death and a δ-finite measureQm such thatY is stationary underQm and Markov with respect to (Pt).For a setB inE the hitting (resp. last exit) time ofY is denoted by τB (resp.λB), andB is called transient (resp. cotransient) ifQm(λB=∞)= 0 (resp.Qm(τB= − ∞)=0. The main theorem then states that for a both transient and contransient setB the distributions ofλB and τB underQm are the same. For suchB the capacity is denfined byC(B):=Qm(λB∈[0, 1] and the cocapacity by