scholarly journals Piecewise deterministic Markov processes and their invariant measures

2021 ◽  
Vol 57 (3) ◽  
Author(s):  
Alain Durmus ◽  
Arnaud Guillin ◽  
Pierre Monmarché
Keyword(s):  
Markov Processes ◽  
Invariant Measures ◽  
Piecewise Deterministic Markov Processes

scholarly journals Random switching near bifurcations

2019 ◽  
Vol 20 (02) ◽  
pp. 2050008 ◽  
Author(s):  
Tobias Hurth ◽  
Christian Kuehn
Keyword(s):  
Markov Processes ◽  
Vector Fields ◽  
Invariant Measures ◽  
Blow Up ◽  
Nonlinear Models ◽  
Switching Dynamics ◽  
Piecewise Deterministic Markov Processes ◽  
Random Switching ◽  
Pitchfork Bifurcations

The interplay between bifurcations and random switching processes of vector fields is studied. More precisely, we provide a classification of piecewise-deterministic Markov processes arising from stochastic switching dynamics near fold, Hopf, transcritical and pitchfork bifurcations. We prove the existence of invariant measures for different switching rates. We also study when the invariant measures are unique, when multiple measures occur, when measures have smooth densities, and under which conditions finite-time blow-up occurs. We demonstrate the applicability of our results for three nonlinear models arising in applications.

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.

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