Summary: We consider a sufficient condition for mild solutions to exist and to be almost surely exponentially stable or exponentially ultimate bounded in mean square for the following stochastic evolution equation with infinite delays driven by Poisson jump processes:
\[ \begin{cases} dX(t) = \left[AX(t)+f(t,X(t-\rho(t)))+\int^0_{-\infty} g(\theta,X(t+\theta))\,d\theta\right]dt\\ \quad\quad +\left[\int^0_{-\infty} h(\theta,X((t+\theta)-))\,d\theta\right]\,dW(t)\\ \quad\quad +\int_U \int^0_{-\infty} k(\theta,X((t+\theta)-),y)\,d\theta q (dtdy),\quad t\geq 0\end{cases} \]
with an initial function \(X(s) =\varphi(s)\), \(-\infty< s\leq 0\), where \(\varphi:(-\infty,0]\to H\) is a càdlàg function with \(E[\sup_{-\infty<s\leq 0}|\varphi(s)|^2_H<\infty\).