Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions. (English) Zbl 1338.60152

Berglund, Nils; Kuehn, Christian

Electron. J. Probab. 21, Paper No. 18, 48 p. (2016); corrigendum ibid. 24, Paper No. 113, 22 p. (2019).

Summary: We prove the 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 a large class of cubic nonlinearities.

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
35R60	PDEs with randomness, stochastic partial differential equations
35K57	Reaction-diffusion equations
81S20	Stochastic quantization
82C28	Dynamic renormalization group methods applied to problems in time-dependent statistical mechanics

Keywords:

stochastic partial differential equations; parabolic equations; reaction-diffusion equations; FitzHugh-Nagumo equation; regularity structures; renormalisation