A1112
Title: Bayesian nonparametrics for semi-linear stochastic PDEs
Authors: Randolf Altmeyer - Imperial College London (United Kingdom) [presenting]
Sascha Gaudlitz - Humboldt-University of Berlin (Germany)
Abstract: The aim is to consider the nonparametric estimation of the reaction function in a semi-linear stochastic partial differential equation (SPDE) from observing a trajectory of the solution continuously over a finite time interval. Given a Gaussian process prior, Bayesian posterior contraction rates are derived in a novel asymptotic regime: The spatial domain grows while the time horizon remains fixed. In this setting, the solution of the SPDE converges to a stationary process and is spatially ergodic. This allows for proving concentration inequalities of functionals along spatial averages of the solution. The proofs rely on the Clark-Ocone formula from Malliavin Calculus and precise bounds on the marginal densities of the SPDE. The posterior contraction rates are minimax-optimal. A nonparametric Bernstein-von Mises theorem is further proven for the posterior distribution.
