scholarly journals A method to compute the transition function of a piecewise deterministic Markov process with application to reliability

2008 ◽  
Vol 78 (12) ◽  
pp. 1397-1403 ◽  
Author(s):  
Julien Chiquet ◽  
Nikolaos Limnios
Keyword(s):  
Markov Process ◽  
Transition Function ◽  
Piecewise Deterministic Markov 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.

Multiple points for symmetric Lévy processes

1978 ◽  
Vol 83 (1) ◽  
pp. 83-90 ◽  
Author(s):  
John Hawkes
Keyword(s):  
Markov Process ◽  
Euclidean Space ◽  
Lévy Process ◽  
Lévy Processes ◽  
Transition Function ◽  
Levy Processes ◽  
Levy Process ◽  
Dimensional Euclidean Space ◽  
Multiple Points ◽  
Image Position

Let Xt be a Lévy process in Rd, d-dimensional euclidean space. That is X is a Markov process whose transition function satisfies

Creation of Mass Processes and Perturbation Theory

1973 ◽  
Vol 25 (3) ◽  
pp. 456-467
Author(s):  
Talma Leviatan
Keyword(s):  
Perturbation Theory ◽  
Markov Process ◽  
Markov Processes ◽  
Transition Function ◽  
Semigroup Of Operators ◽  
Starting Time ◽  
Terminal Time

Creation of mass processes were treated lately by several authors. The idea was to find some generalized Markov process that will correspond to a semigroup of operators which are not necessarily contraction operators (or equivalently to a quasi transition function which is not submarkov). It was G. A. Hunt [6] who first suggested the idea of Markov processes in which both the starting time and the terminal time are random. Such processes were constructed by Helms [4] and treated also by Nagasawa [12] and the author [10].

scholarly journals Subgeometric rates of convergence for a class of continuous-time Markov process

2005 ◽  
Vol 42 (03) ◽  
pp. 698-712
Author(s):  
Zhenting Hou ◽  
Yuanyuan Liu ◽  
Hanjun Zhang
Keyword(s):  
Markov Process ◽  
Continuous Time ◽  
Invariant Probability Measure ◽  
Sufficient Conditions ◽  
Rates Of Convergence ◽  
Transition Function ◽  
Total Variation Norm ◽  
Rate Functions ◽  
General State Space ◽  
Explicit Bounds

Let (Φ t ) t∈ℝ+ be a Harris ergodic continuous-time Markov process on a general state space, with invariant probability measure π. We investigate the rates of convergence of the transition function P t (x, ·) to π; specifically, we find conditions under which r(t)||P t (x, ·) − π|| → 0 as t → ∞, for suitable subgeometric rate functions r(t), where ||·|| denotes the usual total variation norm for a signed measure. We derive sufficient conditions for the convergence to hold, in terms of the existence of suitable points on which the first hitting time moments are bounded. In particular, for stochastically ordered Markov processes, explicit bounds on subgeometric rates of convergence are obtained. These results are illustrated in several examples.

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.

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

2012 ◽  
Vol 44 (03) ◽  
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.

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