Stochastic FitzHugh-Nagumo Equations
U. Böhm (TU Delft - Electrical Engineering, Mathematics and Computer Science)
M.C. Veraar – Mentor (TU Delft - Electrical Engineering, Mathematics and Computer Science)
A. Agresti – Graduation committee member (Ista Nederland B.V. )
G.N.J.C. Bierkens – Graduation committee member (TU Delft - Electrical Engineering, Mathematics and Computer Science)
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Abstract
The stochastic FitzHugh-Nagumo equations are a system of stochastic partial differential equations that describes the propagation of action potentials along nerve axons. In the present work we obtain well-posedness and regularisation results for the FitzHugh-Nagumo equations with domain R^d. We begin by considering the weak critical variational setting, where we prove global well-posedness for the case d=1. We subsequently consider the strong variational setting, which allows us to extend our well-posedness results to d <= 4. To prove well-posedness and regularisation for arbitrary d, we consider the FitzHugh-Nagumo equations in the L^p(L^q)-setting. Building on earlier results for reaction-diffusion equations, we first prove well-posedness on the d-dimensional flat torus and use bootstrapping techniques to prove instantaneous regularisation of the solution. We subsequently extend the theory for reaction-diffusion equations to the unbounded domain R^d to finally prove well-posedness and regularisation for the FitzHugh-Nagumo equations on R^d.