Open Access
2016 Regularity structures and renormalisation of FitzHugh–Nagumo SPDEs in three space dimensions
Nils Berglund, Christian Kuehn
Electron. J. Probab. 21: 1-48 (2016). DOI: 10.1214/16-EJP4371


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 a large class of cubic nonlinearities.


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Nils Berglund. Christian Kuehn. "Regularity structures and renormalisation of FitzHugh–Nagumo SPDEs in three space dimensions." Electron. J. Probab. 21 1 - 48, 2016.


Received: 16 June 2015; Accepted: 18 February 2016; Published: 2016
First available in Project Euclid: 25 February 2016

zbMATH: 1338.60152
MathSciNet: MR3485360
Digital Object Identifier: 10.1214/16-EJP4371

Primary: 35K57 , 60H15
Secondary: 81S20 , 82C28

Keywords: FitzHugh–Nagumo equation , Parabolic equations , reaction–diffusion equations , Regularity structures , renormalisation , Stochastic partial differential equations

Vol.21 • 2016
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