  This work presents a probabilistic scheme for solving semilinear nonlocal
diffusion equations with volume constraints and integrable kernels. The
nonlocal model of interest is defined by a time-dependent semilinear partial
integro-differential equation (PIDE), in which the integro-differential
operator consists of both local convection-diffusion and nonlocal diffusion
operators. Our numerical scheme is based on the direct approximation of the
nonlinear Feynman-Kac formula that establishes a link between nonlinear PIDEs
and stochastic differential equations. The exploitation of the Feynman-Kac
representation successfully avoids solving dense linear systems arising from
nonlocality operators. Compared with existing stochastic approaches, our method
can achieve first-order convergence after balancing the temporal and spatial
discretization errors, which is a significant improvement of existing
probabilistic/stochastic methods for nonlocal diffusion problems. Error
analysis of our numerical scheme is established. The effectiveness of our
approach is shown in two numerical examples. The first example considers a
three-dimensional nonlocal diffusion equation to numerically verify the error
analysis results. The second example presents a physics problem motivated by
the study of heat transport in magnetically confined fusion plasmas.
