## Electronic Journal of Probability

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

#### Abstract

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

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

Dates
Accepted: 18 February 2016
First available in Project Euclid: 25 February 2016

https://projecteuclid.org/euclid.ejp/1456412958

Digital Object Identifier
doi:10.1214/16-EJP4371

Mathematical Reviews number (MathSciNet)
MR3485360

Zentralblatt MATH identifier
1338.60152

#### Citation

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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