Statistical inference for stochastic partial differential equations from local measurements

Stochastic partial differential equations are a multifaceted field where both theoretical and applied problems arise. While there is a rich literature on analytical and probabilistic matters, works on statistical aspects are limited, leaving many research questions unanswered. This dissertation aims to bridge some of these gaps by exploring the statistical potential of the novel local measurement approach.<br/><br/>Paper A is devoted to the joint parameter estimation for coefficients in a linear stochastic convection-diffusion equation. A modified log-likelihood approach leads to an asymptotically normal estimator and the derived central limit theorem generalises previous results. Robustness and applicability of the estimator are discussed. Moreover, minimax rate-optimality, i.e., a lower bound with the same rate of convergence, is established based on innovative insights on the reproducing kernel Hilbert space of the stochastic heat equation and its relation to the Hellinger distance between Gaussian measures.<br/><br/>Paper B examines nonparametric estimation of a spatially varying velocity. The constructed pointwise estimator is motivated through a local log-likelihood approach, and weight functions known from nonparametric regression are introduced. The estimator is decomposed into bias and variance components which are balanced through an additional bandwidth parameter. Under Hölder smoothness conditions, classical nonparametric convergence rates are achieved and their optimality is verified through an adaptation of the lower bounds approach in Paper A. Furthermore, the estimation procedure is extended to both integrated risk and unknown diffusivity level.<br/><br/>Paper C addresses multivariate change estimation for the stochastic heat equation where the discontinuous diffusivity has a jump occurring at some hypersurface. An estimator for the change area is constructed by a CUSUM approach. It consists of the union of optimally chosen pixels. The quality of the estimator is evaluated in terms of the symmetric difference pseudometric. Its analysis depends on the area's underlying complexity, i.e., its boundary roughness, and on the concentration of empirical processes. The results are discussed for the special cases of both graph representation and convexity of the change area.<br/><br/>Paper D focuses on hyperbolic stochastic partial differential equations, and the considered second-order Cauchy problem is given by an elastic system, whose intensity and energy development is characterised by unknown parameters. Combining methods and ideas from both parabolic and hyperbolic equations, a joint central limit theorem for the unknown coefficients is established. The derived convergence rate reflects the impact of the system's damping, i.e., energy loss, as underlying coefficients are more difficult to identify when the magnitude of the damping increases.

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