## Journals

- Advances in Mathematics of Communications
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- Journal of Dynamics & Games
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### Open Access Journals

NHM

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 [21]. 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 [13] to solve a
class of Stochastic Optimal Control Problems inspired by neurological dynamics.

EECT

In the present paper we derive the existence and uniqueness of the solution for the optimal control problem governed by the stochastic FitzHugh-Nagumo equation with recovery variable. Since the drift coefficient is characterized by a cubic non-linearity, standard techniques cannot be applied, instead we exploit the Ekeland's variational principle.

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