## Electronic Journal of Probability

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

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

Nils Berglund and Christian Kuehn

#### 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**

Received: 16 June 2015

Accepted: 18 February 2016

First available in Project Euclid: 25 February 2016

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/16-EJP4371

**Mathematical Reviews number (MathSciNet)**

MR3485360

**Zentralblatt MATH identifier**

1338.60152

**Subjects**

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]

**Keywords**

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

**Rights**

Creative Commons Attribution 4.0 International License.

#### 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