Ergodicity of conditional McKean-Vlasov jump diffusions

In this paper, we are interested in conditional McKean-Vlasov jump diffusions, which are also termed as McKean-Vlasov stochastic differential equations with jump idiosyncratic noise and jump common noise. As far as conditional McKean-Vlasov jump diffusions are concerned, the corresponding conditional distribution flow is a measure-valued process, which indeed satisfies a stochastic partial integral differential equation driven by a Poisson random measure. Via a novel construction of the asymptotic coupling by reflection, we explore the ergodicity of the underlying measure-valued process corresponding to a one-dimensional conditional McKean-Vlasov jump diffusion when the associated drift term fulfils a partially dissipative condition with respect to the spatial variable. In addition, the theory derived demonstrates that the intensity of the jump common noise and the jump idiosyncratic noise can simultaneously enhance the convergence rate of the exponential ergodicity.