  In this paper, we study the problem of Poisson stability of solutions for
stochastic semi-linear evolution equation driven by fractional Brownian motion
$$\mathrm{d} X(t)= \left( AX(t) + f(t,X(t)) \right) \mathrm{d}t +
g\left(t,X(t)\right)\mathrm{d}B^H_{Q}(t),$$ where $A$ is an exponentially
stable linear operator acting on a separable Hilbert space $\mathbb{H}$,
coefficients $f$ and $g$ are Poisson stable in time, and $B^H_Q (t)$ is a
$Q$-cylindrical fBm with Hurst index $H$. First, we establish the existence and
uniqueness of the solution for this equation. Then, we prove that under the
condition where the functions $f$ and $g$ are sufficiently "small", the
equation admits a solution that exhibits the same character of recurrence as
$f$ and $g$. The discussion is further extended to the asymptotic stability of
these Poisson stable solutions. Finally, we include an example to validate our
results.
