EXISTENCE OF STRONG SOLUTIONS FOR NEURONAL NETWORK DYNAMICS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS

Abstract

We study the existence of strong solutions for a class of stochastic differential equations in an infinite dimensional space. Our investigation is specially motivated by the stochastic version of a common model of potential spread in a dendritic tree. We do not assume the noise in the junction points to be Markovian. In fact, we allow for long-range dependence in time of the stochastic perturbation. This leads to an abstract formulation in terms of a stochastic diffusion with dynamic boundary conditions, featuring fractional Brownian motion. We prove results on existence, uniqueness and asymptotics of weak and strong solutions to such a stochastic differential equation.