Pathways into Mathematics of SPDEs:
A Workshop for Young Researchers

The statistical analysis of the stochastic wave equation requires analytical tools that differ from those used for parabolic SPDEs; one such tool is the Riemann–Lebesgue lemma. Using prototypical examples, we will show how the Riemann–Lebesgue lemma arises in the classical deterministic theory for the wave equation, in analytical properties of the stochastic wave equation, and in the statistical analysis of three major observation schemes: the spectral, local, and discrete. In particular, we will illustrate connections between energy and information, as well as with classical Fourier-analytic concepts such as Cesàro summation and Fejér kernels.

We study spectral observations for Lévy-driven linear parabolic SPDEs with temporally independent jumps of finite or infinite activity. Relying on orthogonal projections onto finite-dimensional subspaces and under the assumption of a non-vanishing white noise component, we derive maximum likelihood-based estimators and discuss their feasibility in both continuous- and discrete-time observation schemes. A key difficulty arises from the presence of the continuous local martingale part of the projected solution process, which enters the likelihood function and obstructs direct implementation. We propose methods to overcome this issue; in the discrete-time setup, we arrive at sufficient conditions for constructing a spectral jump filter in the space-time SPDE framework. Under suitable assumptions, we establish central limit theorems for both continuous- and discrete-time estimators. As the number of observed Fourier modes and – in the discrete case – the number of time observation points simultaneously tend to infinity, the estimators achieve the benchmark convergence rate known from the pure white noise setting.

We consider elliptic random partial differential equations (PDEs), which are solved using the stochastic Galerkin method. This approach leads to high-dimensional coupled systems of deterministic PDEs. Consequently, when using classical numerical solvers, the complexity of applications is usually limited to lower-dimensional problems or special cases where decoupling schemes can be applied. In our approach, we replace the classical numerical solver with deep learning methods. We formulate and compare two different training strategies for the neural networks, based on the strong and weak formulations of the random PDE. The strategy based on the strong formulation is a type of physics-informed neural network (PINN), which is widely used in the literature. The strategy based on the weak formulation minimizes the Ritz energy functional, as introduced as the deep Ritz method. In this training strategy, the residual is of a lower differentiation order, reducing the training cost considerably.

The field of Geology is concerned with the history of the Earth and the processes that shaped it. These processes act in space and time and are naturally modelled by (stochastic) partial differential equations. We discuss derivations and applications of well-known (S)PDEs, e.g. diffusion and advection equations, in landscape evolution models. Additionally, we present some insights on research questions, practical aspects and challenges from an application perspective.