scholarly journals On absolute continuity of invariant measures associated with a piecewise-deterministic Markov process with random switching between flows

2021 ◽  
Vol 213 ◽  
pp. 112522
Author(s):  
Dawid Czapla ◽  
Katarzyna Horbacz ◽  
Hanna Wojewódka-Ściążko
Keyword(s):  
Markov Process ◽  
Absolute Continuity ◽  
Invariant Measures ◽  
Piecewise Deterministic Markov Process ◽  
Random Switching

scholarly journals Randomly switched vector fields sharing a zero on a common invariant face

2020 ◽  
Vol 21 (02) ◽  
pp. 2150007
Author(s):  
Edouard Strickler
Keyword(s):  
Dynamical Systems ◽  
Markov Process ◽  
Lyapunov Exponents ◽  
Vector Fields ◽  
Degenerate Case ◽  
Piecewise Deterministic Markov Process ◽  
Random Switching ◽  
Invariant Densities

We consider a Piecewise Deterministic Markov Process given by random switching between finitely many vector fields vanishing at [Formula: see text]. It has been shown recently that the behavior of this process is mainly determined by the signs of Lyapunov exponents. However, results have only been given when all these exponents have the same sign. In this paper, we consider the degenerate case where the process leaves invariant some face and results are stated when the Lyapunov exponents are of opposite signs. Our results enable in particular to close a gap in a discussion on random switching between two Lorenz vector fields by Bakhtin and Hurth, Invariant densities for dynamical systems with random switching, Nonlinearity 25 (2012) 2937–2952.

scholarly journals On invariant measures of nonlinear Markov processes

1993 ◽  
Vol 6 (4) ◽  
pp. 385-406 ◽  
Author(s):  
N. U. Ahmed ◽  
Xinhong Ding
Keyword(s):  
Markov Process ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Markov Processes ◽  
Existence And Uniqueness ◽  
Invariant Measures ◽  
Nonlinear Markov Processes

We consider a nonlinear (in the sense of McKean) Markov process described by a stochastic differential equations in Rd. We prove the existence and uniqueness of invariant measures of such process.

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.

The calculation of limit probabilities for denumerable Markov processes from infinitesimal properties

1973 ◽  
Vol 10 (01) ◽  
pp. 84-99 ◽  
Author(s):  
Richard L. Tweedie
Keyword(s):  
Markov Process ◽  
Continuous Time ◽  
Transition Matrix ◽  
Invariant Measures ◽  
Equilibrium Distribution ◽  
Transition Functions ◽  
The Matrix ◽  
Limit Probability ◽  
Derivatives Of ◽  
Regular Chains

The problem considered is that of estimating the limit probability distribution (equilibrium distribution) πof a denumerable continuous time Markov process using only the matrix Q of derivatives of transition functions at the origin. We utilise relationships between the limit vector πand invariant measures for the jump-chain of the process (whose transition matrix we write P∗), and apply truncation theorems from Tweedie (1971) to P∗. When Q is regular, we derive algorithms for estimating πfrom truncations of Q; these extend results in Tweedie (1971), Section 4, from q-bounded processes to arbitrary regular processes. Finally, we show that this method can be extended even to non-regular chains of a certain type.

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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