On the impact of noise on quenching for a nonlocal diffusion model driven by a mixture of Brownian and fractional Brownian motions

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 is driven by a mixture of a classical Brownian and a fractional Brownian motion with Hurst index $ H\in(\frac{1}{2}, 1). $ We first establish local in time existence results and then explore conditions under which the resulting SPDE exhibits finite-time quenching. Using results on the probability distribution of perpetual integral functionals of Brownian motion as well as tail estimates for the fractional Brownian motion we provide analytic estimates for certain quantities of interest, such as upper bounds for 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 noise on the dynamics of the system. The analytic results are complemented with a detailed numerical study of the model under Dirichlet boundary conditions. A possible application concerning MEMS technology is considered and the implications of the results in this context are commented upon.