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

scholarly journals Averaging for a Fully Coupled Piecewise-Deterministic Markov Process in Infinite Dimensions

2012 ◽  
Vol 44 (3) ◽  
pp. 749-773 ◽  
Author(s):  
Alexandre Genadot ◽  
Michèle Thieullen
Keyword(s):  
Ion Channels ◽  
Markov Process ◽  
Time Scale ◽  
Evolution Equations ◽  
Scale Model ◽  
Infinite Dimensions ◽  
Fast Analysis ◽  
Piecewise Deterministic Markov Process ◽  
Averaged Model ◽  
Fully Coupled

In this paper we consider the generalized Hodgkin-Huxley model introduced in Austin (2008). This model describes the propagation of an action potential along the axon of a neuron at the scale of ion channels. Mathematically, this model is a fully coupled piecewise-deterministic Markov process (PDMP) in infinite dimensions. We introduce two time scales in this model in considering that some ion channels open and close at faster jump rates than others. We perform a slow-fast analysis of this model and prove that, asymptotically, this ‘two-time-scale’ model reduces to the so-called averaged model, which is still a PDMP in infinite dimensions, for which we provide effective evolution equations and jump rates.

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.

Global dynamical behavior of a multigroup SVIR epidemic model with Markovian switching

2021 ◽  
pp. 2150080
Author(s):  
Qun Liu ◽  
Daqing Jiang
Keyword(s):  
Markov Process ◽  
Lyapunov Function ◽  
Stochastic System ◽  
Epidemic Model ◽  
Regime Switching ◽  
Dynamical Behavior ◽  
Piecewise Deterministic Markov Process ◽  
Positive Recurrence ◽  
Persistence In The Mean ◽  
The Mean

In this paper, we are concerned with the global dynamical behavior of a multigroup SVIR epidemic model, which is formulated as a piecewise-deterministic Markov process. We first obtain sufficient criteria for extinction of the diseases. Then we establish sufficient criteria for persistence in the mean of the diseases. Moreover, in the case of persistence, we find a domain which is positive recurrence for the solution of the stochastic system by constructing an appropriate Lyapunov function with regime switching.

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.

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