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