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From diffusion to reaction - Statistical Inference for semi-linear SPDEs

Sascha Gaudlitz (Humboldt-Universität zu Berlin)
Thursday 19 May 2022 13:15–14:00 Aud. D4 (1531-219)
Stochastics Seminar

Stochastic partial differential equations (SPDEs) form a convenient class of mathematical models for random spatio-temporal dynamics. Of particular interest are stochastic reaction-diffusion equations, or more generally, semi-linear SPDEs. In this talk, I will first give a brief introduction to semi-linear SPDEs and the statistical questions related to them. Subsequently, an estimator for the reaction intensity will be deduced. Consistent inference is achieved by studying a small diffusivity level, which is realistic in applications. The main result is a central limit theorem for the estimation error of a parametric estimator, from which confidence intervals can be constructed. Statistical efficiency is demonstrated by establishing local asymptotic normality. Local observations allow for non-parametric estimation of a reaction intensity varying in time and space. The statistical analysis requires advanced tools from stochastic analysis like Malliavin calculus for SPDEs, the infinite-dimensional Gaussian Poincaré inequality and regularity results for SPDEs in $L^p$-interpolation spaces.

This is joint work with Markus Reiß.

Organised by: Stochastics Group
Contact: Claudia Strauch Revised: 18.08.2022