Green measures for Markov processes (original) (raw)

Green Measures for Time Changed Markov Processes

2020

In this paper we study Green measures for certain classes of random time change Markov processes where the random time change are inverse subordinators. We show the existence of the Green measure for these processes under the condition of the existence of the Green measure of the original Markov processes and they coincide. Applications to fractional dynamics in given.

On the Green functions of {1}-dimensional diffusion processes

Publications of the Research Institute for Mathematical Sciences, 1980

On the Green Functions of 1-DImensIona! Diffusion Processes By Yuji KASAHARA, Shin'ichi KOTANI and Hisao WATANABE §0. Introduction Let X= {X t } be a Markov process with a measurable state S. The Green operator G a (a^O) of Xis usually defined by ft f(x \ _ w r /(X) A,l = (<?"(*, cfy)/(j) Js for every bounded measurable function/(x), where G.(x, E) = For every a>0, G# is always well-defined, but G 0 does not exist in general, as is seen in the case of 1-or 2-dimensional Brownian motions. However, in many cases, G 0 /can be defined to be the limit of G^/as a-»0 if we restrict /to be in a certain class of functions, called the domain of the potential operator. There have been a lot of works on potential operators for Markov processes, e.g.

On the Conservativeness of Some Markov Processes

Potential Analysis, 2016

We give a unified method to obtain the conservativeness of a class of Markov processes associated with lower bounded semi-Dirichlet forms on L 2 (X; m), including symmetric diffusion processes, some non-symmetric diffusion processes and jump type Markov processes on X, where X is a locally compact separable metric space and m is a positive Radon measure on X with full topological support. Using the method, we give an example in each section, providing the conservativeness of the processes, that are given by the "increasingness of the volume of some sets(balls)" and "that of the coefficients on the sets" of the Markov processes.

Martingales on Jump Processes. I: Representation Results

SIAM Journal on Control, 1975

The paper is a contribution to the theory of martingales of processes whose sample paths are piecewise constant and have finitely many discontinuities in a finite time interval. The assumption is made that the jump times of the underlying process are totally inaccessible and necessary and sufficient conditions are given for this to be true. It turns out that all martingales are then discontinuous, and can be represented as stochastic integrals of certain basic martingales. This representation theorem is used in a companion paper to study various practical problems in communication and control. The results in the two papers constitute a sweeping generalization of recent work on Poisson processes.

A technique for exponential change of measure for Markov processes

2002

We consider a Markov process X (t) with extended generator A and domain D(A). Let fF t g be a right-continuous history filtration and P t denote the restriction of P to F t. LetP P be another probability measure on (Ù, F) such that dP P t =dP t ¼ E h (t), where E h (t) ¼ h(X (t)) h(X (0)) exp À ð t 0 (Ah)(X (s)) h(X (s)) ds is a true martingale for a positive function h 2 D(A). We demonstrate that the process X (t) is a Markov process on the probability space (Ù, F , fF t g,P P), we find its extended generatorà A and provide sufficient conditions under which D(à A) ¼ D(A). We apply this result to continuous-time Markov chains, to piecewise deterministic Markov processes and to diffusion processes (in this case a special choice of h yields the classical Cameron-Martin-Girsanov theorem).

On solutions of Kolmogorov's equations for jump Markov processes

arXiv (Cornell University), 2013

This paper studies three ways to construct a nonhomogeneous jump Markov process: (i) via a compensator of the random measure of a multivariate point process, (ii) as a minimal solution of the backward Kolmogorov equation, and (iii) as a minimal solution of the forward Kolmogorov equation. The main conclusion of this paper is that, for a given measurable transition intensity, commonly called a Q-function, all these constructions define the same transition function. If this transition function is regular, that is, the probability of accumulation of jumps is zero, then this transition function is the unique solution of the backward and forward Kolmogorov equations. For continuous Q-functions, Kolmogorov equations were studied in Feller's seminal paper. In particular, this paper extends Feller's results for continuous Q-functions to measurable Q-functions and provides additional results.

Heat kernel bounds for a large class of Markov process with singular jump

Stochastic Processes and their Applications, 2021

Let Z = (Z 1 ,. .. , Z d) be the d-dimensional Lévy processes where Z i 's are independent 1-dimensional Lévy processes with jump kernel J φ,1 (u, w) = |u−w| −1 φ(|u− w|) −1 for u, w ∈ R. Here φ is an increasing function with weak scaling condition of order α, α ∈ (0, 2). Let J(x, y) ≍ J φ (x, y) be the symmetric measurable function where J φ (x, y) := J φ,1 (x i , y i) if x i = y i for some i and x j = y j for all j = i 0 if x i = y i for more than one index i. Corresponding to the jump kernel J, we show the existence of non-isotropic Markov processes X := (X 1 ,. .. , X d) and obtain two-sided heat kernel estimates for the transition density functions, which turn out to be comparable to that of Z.

The technique of the exponential change of measure for Markov processes

Bernoulli

We consider Markov process fX(t); t 0g with the extended generator A and the domain D(A). Let fF t g be the right-continuous history filtration and by IP t we denote the restriction of IP to fF t g. Let ~ IP be another probability measure on(Omega ; F) such that d ~ IP t =dIP t = M (t), where M (t) = h(X(t)) h(X(0)) exp ` Gamma Z t 0 (Ah)(X(s)) h(X(s)) ds ' ; is a true martingale for a positive function h 2 D(A). In this note we demonstrate that process fX(t); t 0g is a Markov process on the probability space(Omega ; F ; ~ IP) and we show its extended generator ~ A. We apply this result to CTMC, PDMP and to diffusion processes (in this case a special choice of h yields the classical CameronMartin -Girsanov Theorem). Keywords: Markov process, extended generator, exponential change of measure, CameronMartin -Girsanov theorem, (local) martingale. AMS 1991 Subject Classification: Primary: 60G44, 60G25, 60J25, 60J35; Secondary: 60G48, 60H05, 60J57. This work was partiall...

Donsker-Varadhan asymptotics for degenerate jump Markov processes

We consider a class of continuous time Markov chains on a compact metric space that admit an invariant measure strictly positive on open sets together with absorbing states. We prove the joint large deviation principle for the empirical measure and flow. Due to the lack of uniform ergodicity, the zero level set of the rate function is not a singleton. As corollaries, we obtain the Donsker-Varadhan rate function for the empirical measure and a variational expression of the rate function for the empirical flow.