Conference Agenda

We consider the Bayesian non-parametric estimation of the reaction term in a semi-linear parabolic SPDE. Consistency is achieved by making use of the spatial ergodicity of the SPDE while the time horizon is fixed. The analysis of the estimation error requires new concentration results for spatial averages of transformation of the SPDE, which are based the combination of the Clark-Ocone formula with bounds on the marginal densities. The general methodology is exemplified in the asymptotic regime, where the diffusivity level and the noise level of the SPDE tend to zero in a realistic coupling.

We estimate the diffusivity (or drift) in the stochastic Burgers equation driven by additive space-time white noise. Our estimator is based on the local measurements, i.e., we assume that the solution is measured locally in space and over a finite time interval. Such estimator has been introduced in [2] for linear SPDEs with additive noise, but [3] considered also the multiplicative noise case and [1] applied it to a large class of semilinear SPDEs, namely to the stochastic Burgers equation with "smooth" noise. Our work contributes to the topic and extends achieved results.

In our talk, we first assert the regularity of both the linear part of the solution (i.e., the stochastic convolution) and the nonlinear part. Then we show that our proposed estimator is strong consistent and asymptotically normal.

This is a joint work with Professor Enrico Priola.

References:

[1] Altmeyer, R., Cialenco, I., Pasemann, G., (2023): Parameter estimation for semilinear SPDEs from local measurements. Bernoulli 29(3), 2035–2061.

[2] Altmeyer, R., Reiss, M., (2021): Nonparametric estimation for linear SPDEs from local measurements. Annals of Applied Probability 31(1), 1–38.

[3] Janak, J., Reiss, M., (2024): Parameter estimation for the stochastic heat equation with multiplicative noise from local measurements. Stochastic Processes and their Applications 175.

The coefficients of elastic and dissipative operators in a linear hyperbolic SPDE are jointly estimated using multiple spatially localised measurements. As the resolution level of the observations tends to zero, we establish the asymptotic normality of an augmented maximum likelihood estimator. The rate of convergence for the dissipative coefficients matches rates in related parabolic problems, whereas the rate for the elastic parameters also depends on the magnitude of the damping. The analysis of the observed Fisher information matrix relies upon the asymptotic behaviour of rescaled M,N-functions generalising the operator sine and cosine families appearing in the undamped wave equation. In the undamped case, the observed Fisher information is intrinsically related to the kinetic energy within a deterministic wave equation and the notion of Riemann-Lebesgue operators.