Partial differential equations (PDE) are ubiquitous in describing real-world phenomena. In many statistical models, PDE are used to encode complex relationships between unknown quantities and the observed data. We investigate statistical and computational questions arising in such models, adopting an infinite-dimensional `nonparametric' framework and assuming the observed data are subject to random noise. The main PDE examples are of elliptic or parabolic type.

Chapter 2 investigates the problem of sampling from high-dimensional Bayesian posterior distributions. The main results consist of non-asymptotic computational guarantees for Langevin-type Markov chain Monte Carlo (MCMC) algorithms which scale polynomially in key quantities such as the dimension of the model, the desired precision level, and the number of available statistical measurements. The bounds hold with high probability under the distribution of the data, assuming that certain `local geometric' assumptions are fulfilled and that a good initialiser of the algorithm is available. We study a representative non-linear PDE example where the unknown is a coefficient function in a steady-state Schr"odinger equation, and the solution to a corresponding boundary value problem is observed.

Chapter 3 studies statistical convergence rates for nonparametric Tikhonov-type estimators, which can be interpreted also as Bayesian maximum a posteriori (MAP) estimators arising from certain Gaussian process priors. The theory is derived in a general setting for non-linear inverse problems and then applied to two examples, the steady-state Schr"odinger equation studied in Chapter \ref{sampling} and a model for the steady-state heat equation. It is shown that the rates obtained are minimax-optimal in prediction loss.

The final Chapter 4 considers a model for scalar diffusion processes $(Xt:t\ge 0)$ with an unknown drift function which is modelled nonparametrically. It is shown that in the low frequency sampling case, when the sample consists of $(X0,X\Delta ,...,Xn\Delta )$ for some fixed sampling distance $\Delta >0$, under mild regularity assumptions, the model satisfies the local asymptotic normality (LAN) property. The key tools used are regularity estimates and spectral properties for certain parabolic and elliptic PDE related to $(Xt:t\ge 0)$.