Convergence to equilibrium for time inhomogeneous jump diffusions with state dependent jump intensity
Abstract
We consider a time inhomogeneous jump Markov process with state dependent jump intensity, taking values in Its infinitesimal generator is given by \begin{multline*} L_t f (x) = \sum_{i=1}^d \frac{\partial f}{\partial x_i } (x) b^i ( t,x) - \sum_{ i =1}^d \frac{\partial f}{\partial x_i } (x) \int_{E_1} c_1^i ( t, z, x) \gamma_1 ( t, z, x ) \mu_1 (dz ) \\ + \sum_{l=1}^3 \int_{E_l} [ f ( x + c_l ( t, z, x)) - f(x)] \gamma_l ( t, z, x) \mu_l (dz ) , \end{multline*} where are sigma-finite measurable spaces describing three different jump regimes of the process (fast, intermediate, slow). We give conditions proving that the long time behavior of can be related to the one of a time homogeneous limit process Moreover, we introduce a coupling method for the limit process which is entirely based on certain of its big jumps and which relies on the regeneration method. We state explicit conditions in terms of the coefficients of the process allowing to control the speed of convergence to equilibrium both for and for
Cite
@article{arxiv.1712.03507,
title = {Convergence to equilibrium for time inhomogeneous jump diffusions with state dependent jump intensity},
author = {Eva Löcherbach},
journal= {arXiv preprint arXiv:1712.03507},
year = {2018}
}