Electronic Journal of Probability

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

Nils Berglund and Christian Kuehn

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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.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 18, 48 pp.

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

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Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 35K57: Reaction-diffusion equations
Secondary: 81S20: Stochastic quantization 82C28: Dynamic renormalization group methods [See also 81T17]

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

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Berglund, Nils; Kuehn, Christian. Regularity structures and renormalisation of FitzHugh–Nagumo SPDEs in three space dimensions. Electron. J. Probab. 21 (2016), paper no. 18, 48 pp. doi:10.1214/16-EJP4371. https://projecteuclid.org/euclid.ejp/1456412958

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