Abstract: We study a class of reaction-diffusion type equations on a finite network with continuity assumptions and a kind of non-local, stationary Kirchhoff's conditions at the nodes. A multiplicative random Gaussian perturbation acting along the edges is also included. For such a problem we prove Gaussian estimates for the semigroup generated by the evolution operator, hence generalizing similar results previously obtained in . In particular our main goal is to extend known results on Gaussian upper bounds for heat equations on networks with local boundary conditions to those with non-local ones. We conclude showing how our results can be used to apply techniques developed in  to solve a class of Stochastic Optimal Control Problems inspired by neurological dynamics.
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