Stochastic partial differential equations (SPDEs) are mathematical models that describe the evolution of dynamical systems in space and time under the influence of random noise. The noise may represent the system’s inherent stochasticity, model uncertainties, or extrinsic stochastic forces acting on the system. By marrying the deterministic dynamics of classical partial differential equations with the influence of random noise, SPDEs allow for modeling of uncertainty in complex spatio-temporal phenomena in a wide range of fields, including geophysics, neuroscience, and finance.

In many real-world applications, a process modelled by an SPDE is observed only discretely and partially. As an example, consider an evolving firefront, whose fire intensity may be observed through satellite sensors at discrete points in time at a fixed spatial resolution. The SPDE model, jointly with the observed data, defines two statistical problems. Firstly, there is the estimation of the latent state of the system, given the observations. Typically, this is separated into two subproblems: the filtering problem, dealing with online state estimation as new observations become available, and the smoothing problem, dealing with offline state estimation given a full set of observations over a fixed time interval. Secondly, there is the question of parameter estimation. Typically, any SPDE model is governed by a set of parameters. These might, for example, represent physical constants that determine the dynamics of the system. In applications where such parameters are unknown, they are to be estimated jointly with the unobserved signal, based on the observed data.

This thesis develops Bayesian computational approaches to statistical inference for SPDEs. Both state estimation, in the online and online setting, as well as parameter estimation for discretely and partially observed semilinear SPDEs are addressed. It therefore fills a gap in the current literature on statistics for SPDEs, which has so far focused primarily on frequentist parameter estimation or state estimation of linear SPDEs. By introducing methodology that enables inference of the latent state, model calibration and uncertainty quantification for semilinear SPDEs, based on incomplete and noisy data, this work broadens the applicability of SPDEs in real-world settings.

In Chapter 2, a class of exponential measure transformations for SPDEs is introduced. Conditions are derived under which the transformed measure is of Girsanov-type such that the mild solution to the SPDE evolves according to yet another SPDE with an additional drift term. An application this result gives rise to the infinite-dimensional diffusion bridge - a mild solution to an SPDE conditioned on hitting a predefined terminal state. This generalises results previously known only for linear systems. Moreover, the guided process, the mild solution to a tractable SPDE that steers a process towards a terminal state, is introduced as an approximation to the diffusion bridge.

Chapter 3 develops sampling methodology for the intractable infinite-dimensional diffusion bridge. This serves as a fundamental building block for computational Bayesian approaches to inference for discretely observed SPDEs. In the main result of Chapter 3, conditions are derived under which absolute continuity holds between the laws of the guided process and the diffusion bridge. This legitimises the guided process as a proposal distribution for importance sampling or Metropolis-Hastings schemes that target the law of the infinite-dimensional diffusion bridge.

In Chapter 4, these ideas are extended to build methodology for all of the aforementioned statistical tasks. To address the filtering problem, a sequential Monte Carlo scheme is introduced, built upon the law of the guided process between observation times as a proposal distribution. The smoothing and parameter inference problems are solved by generalising the measure transformations of Chapter 2 to include conditioning and guiding based on multiple observations. Building on these transformations and a reparametrisation of the conditioned process, a Gibbs sampler is derived that samples from the joint posterior of the smoothed process and unknown model parameters.

Given a mild solution X to a semilinear stochastic partial differential equation (SPDE), we consider an exponential change of measure based on its infinitesimal generator L, defined in the topology of bounded pointwise convergence. The changed measure P^h depends on the choice of a function h in the domain of L. In our main result, we derive conditions on h for which the change of measure is of Girsanov-type. The process X under P^h is then shown to be a mild solution to another SPDE with an extra additive drift-term. We illustrate how different choices of h impact the law of X under P^h in selected applications. These include the derivation of an infinite-dimensional diffusion bridge as well as the introduction of guided processes for SPDEs, generalizing results known for finite-dimensional diffusion processes to the infinite-dimensional case.