Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions

Nils Berglund (1, 2) , Christian Kuehn (3)

1 FDP - Fédération de recherche Denis Poisson
2 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
3 Institute for Analysis and Scientific Computing [Wien]

We prove local existence of solutions for a class of suitably renormalised coupled SPDE–ODE systems driven by space-time white noise, where the space dimension is equal to 2 or 3. This class includes in particular the FitzHugh–Nagumo system describing the evolution of action potentials of a large population of neurons, as well as models with multidimensional gating variables. The proof relies on the theory of regularity structures recently developed by M. Hairer, which is extended to include situations with semigroups that are not regularising in space. We also provide explicit expressions for the renormalisation constants, for general cubic nonlinearities.

Mots clés: FitzHugh–Nagumo equation; reaction–diffusion equations; renormalisation; regularity structures; stochastic partial differential equations; parabolic equations

Domaines: Probabilités [math.PR]

Nils Berglund, Christian Kuehn. Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions. 2015. ⟨hal-01141380v1⟩