Albeverio, Sergio and Di Persio, Luca and Mastrogiacomo, Elisa (2010) *Small noise asymptotic expansions for stochastic pde's.i : The case of a dissipative polynomially bounded non linearity.* UNSPECIFIED. (Submitted)

## Abstract

We study a reaction-diusion evolution equation perturbed by a Gaussian noise. Here the leading operator is the innitesimal generator of a C0-semigroup of strictly negative type, the nonlinear term has at most polynomial growth and is such that the whole system is dissipative. The corresponding It^o stochastic equation describes a process on a Hilbert space with dissipative nonlinear drift and a Gaussian noise. Under smoothness assumptions on the non-linearity, asymptotics to all orders in a small parameter in front of the noise are given, with uniform estimates on the remainders. Applications to nonlinear SPDEs with a linear term in the drift given by a Laplacian in a bounded domain are included. As a particular example we consider the small noise asymptotic expansions for the stochastic FitzHugh-Nagumo equations of neurobiology around deterministic solutions. 2010 Mathematics Subject Classication. Primary 35K57 , 35R60, 35C20 ; Secondary 92B20

Item Type: | Departmental Technical Report |

Department or Research center: | Mathematics |

Subjects: | Q Science > QA Mathematics > QA273 Probabilities |

Uncontrolled Keywords: | Reaction diusion equations, dissipative systems, asymptotic expansions, polynomially bounded non linearity, stochastic FitzHugh-Nagumo system |

Report Number: | UTM 741 |

Repository staff approval on: | 08 Nov 2010 |
---|

### Actions (login required)