Convergence to Equilibrium for Time-Inhomogeneous Jump Diffusions with State-Dependent Jump Intensity

We consider a time-inhomogeneous Markov process $$X = (X_t)_t$$ X = ( X t ) t with jumps having state-dependent jump intensity, with values in $${\mathbb {R}}^d , $$ R d , and we are interested in its longtime behavior. The infinitesimal generator of the process is given for any sufficiently smooth test function f by $$\begin{aligned} L_t f (x) = \sum _{i=1}^d \frac{\partial f}{\partial x_i } (x) b^i ( t,x) + \int _{{\mathbb {R}}^m } [ f ( x + c ( t, z, x)) - f(x)] \gamma ( t, z, x) \mu (\mathrm{d}z ) , \end{aligned}$$ L t f ( x ) = ∑ i = 1 d ∂ f ∂ x i ( x ) b i ( t , x ) + ∫ R m [ f ( x + c ( t , z , x ) ) - f ( x ) ] γ ( t , z , x ) μ ( d z ) , where $$ \mu $$ μ is a $$\sigma $$ σ -finite measure on $$({\mathbb {R}}^m , {\mathcal B} ( {\mathbb {R}}^m ) ) $$ ( R m , B ( R m ) ) describing the jumps of the process. We give conditions on the coefficients b(t, x) , c(t, z, x) and $$ \gamma ( t, z, x ) $$ γ ( t , z , x ) under which the longtime behavior of X can be related to the longtime behavior of a time-homogeneous limit process $${\bar{X}} . $$ X ¯ . 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 control of the speed of convergence to equilibrium both for X and for $${\bar{X}}$$ X ¯ .