Exponential ergodicity for diffusions with jumps driven by a Hawkes process

In this paper, we introduce a new class of processes which are diffusions with jumps driven by multivariate nonlinear Hawkes process. Our goal is to study their long-time behavior. the case exponential memory kernels for underlying process establish conditions positive Harris recurrence couple $(X,Y )$, where $X$ denotes diffusion and $Y$ piecewise deterministic Markov (PDMP) defining stochastic intensity driving Hawkes. As direct consequence recurrence, obtain ergodic theorem $X.$ Furthermore, provide sufficient under exponentially $\beta -$mixing.