  In this paper, we study a stochastic parabolic problem involving a nonlocal
diffusion operator associated with nonlocal Robin-type boundary conditions. The
stochastic dynamics under consideration are driven by a mixture of a classical
Brownian and a fractional Brownian motions with Hurst index $H\in(\frac{1}{2},
1).$ We first establish local in existence result of the considered model and
then explore conditions under which the resulting SPDE exhibits finite-time
quenching. Using the probability distribution of perpetual integral functional
of Brownian motion as well as tail estimates of fractional Brownian motion we
provide analytic estimates for certain statistics of interest, such as
quenching times and the corresponding quenching probabilities. The existence of
global in time solutions is also investigated and as a consequence a lower
estimate of the quenching time is also derived. Our analytical results
demonstrate the non-trivial impact of the considered noise on the dynamics of
the system. Next, a connection of a special case of the examined model is drawn
in the context of MEMS technology. Finally, a numerical investigation of the
considered model for a fractional Laplacian diffusion and Dirichlet-type
boundary conditions is delivered.
