An L p Theory for a Markov Process with a Sub-Invariant Measure

1965 ◽  
Vol 16 (3) ◽  
pp. 398 ◽  
Author(s):  
S. R. Foguel
Keyword(s):  
Markov Process ◽  
Invariant Measure

On Null-Recurrent Markov Chains

1960 ◽  
Vol 12 ◽  
pp. 278-288 ◽  
Author(s):  
John Lamperti
Keyword(s):  
Markov Chain ◽  
Markov Process ◽  
Invariant Measure ◽  
Discrete Time ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Positive Vector ◽  
Scalar Multiple ◽  
Basic Facts ◽  
Image Position

Throughout this paper, the symbol P = [Pij] will represent the transition probability matrix of an irreducible, null-recurrent Markov process in discrete time. Explanation of this terminology and basic facts about such chains may be found in (6, ch. 15). It is known (3) that for each such matrix P there is a unique (except for a positive scalar multiple) positive vector Q = {qi} such that QP = Q, or1this vector is often called the "invariant measure" of the Markov chain.The first problem to be considered in this paper is that of determining for which vectors U(0) = {μi(0)} the vectors U(n) converge, or are summable, to the invariant measure Q, where U(n) = U(0)Pn has components2In § 2, this problem is attacked for general P. The main result is a negative one, and shows how to form U(0) for which U(n) will not be (termwise) Abel summable.

VALIDATING THE DIFFUSION APPROXIMATION THROUGH CONDITIONAL ENTROPIES

2011 ◽  
Vol 25 (06) ◽  
pp. 377-383
Author(s):  
J.-P. RIVET ◽  
F. DEBBASCH
Keyword(s):  
Markov Process ◽  
Invariant Measure ◽  
Diffusion Approximation ◽  
Particle Transport ◽  
Invariant Measures ◽  
Large Range ◽  
Brownian Particle ◽  
Conditional Entropy ◽  
The Real ◽  
Transport Dynamics

The diffusion approximation replaces a real transport dynamics by an approximate stochastic Markov process. It is proposed that, when both dynamics have invariant measures, the conditional entropy of the invariant measure of the real dynamics with respect to the invariant measure of the Markov process be used to assess quantitatively the validity of the approximation. This proposal is tested on particle transport; the diffusion approximation is found to be quite robust, valid for an unexpectedly large range of mass ratios between the solvent and the Brownian particle.

scholarly journals Time Reversions of Markov Processes

1964 ◽  
Vol 24 ◽  
pp. 177-204 ◽  
Author(s):  
Masao Nagasawa
Keyword(s):  
Markov Process ◽  
Invariant Measure ◽  
Markov Chains ◽  
Markov Processes ◽  
Transition Probability ◽  
Initial Distribution ◽  
Invariant Distribution

A time reversion of a Markov process was discussed by Kolmogoroff for Markov chains in 1936 [6] and for a diffusion in 1937 [7l He described it as a process having an adjoint transition probability. Although his treatment is purely analytical, in his case if the process xt has an invariant distribution, the reversed process zt = x-t is the process with the adjoint transition probability. In this discussion, however, it is very restrictive that the initial distribution of the process must be an invariant measure.

scholarly journals Subgeometric rates of convergence for Markov processes under subordination

2017 ◽  
Vol 49 (1) ◽  
pp. 162-181 ◽  
Author(s):  
Chang-Song Deng ◽  
René L. Schilling ◽  
Yan-Hong Song
Keyword(s):  
Markov Process ◽  
Invariant Measure ◽  
Convergence Rate ◽  
Rate Of Convergence ◽  
Rates Of Convergence ◽  
Bernstein Function ◽  
Convergence To Equilibrium ◽  
Speed Of Convergence ◽  
Crucial Point ◽  
Original Process

Abstract We are interested in the rate of convergence of a subordinate Markov process to its invariant measure. Given a subordinator and the corresponding Bernstein function (Laplace exponent), we characterize the convergence rate of the subordinate Markov process; the key ingredients are the rate of convergence of the original process and the (inverse of the) Bernstein function. At a technical level, the crucial point is to bound three types of moment (subexponential, algebraic, and logarithmic) for subordinators as time t tends to ∞. We also discuss some concrete models and we show that subordination can dramatically change the speed of convergence to equilibrium.

scholarly journals Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process

2020 ◽  
Vol 17 (2) ◽  
pp. 1059-1073
Author(s):  
Dawid Czapla ◽  
◽  
Sander C. Hille ◽  
Katarzyna Horbacz ◽  
Hanna Wojewódka-Ściążko ◽  
...  
Keyword(s):  
Markov Process ◽  
Invariant Measure ◽  
Continuous Dependence ◽  
Piecewise Deterministic Markov Process ◽  
Jump Rate
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