Archive

Abstract: Unraveling the general dynamics of nonlinear PDEs is often a grand challenge. Adopting the classical dynamical systems viewpoint, one can follow the strategy of first concentrating on existence, stability and bifurcations of special solutions and then examine their role for more complicated dynamics. Localized structures such as front solutions, pulses and wave packets are precisely such special solutions for infinite-dimensional dynamical systems given by PDEs. In this talk we focus on the dynamics of front solutions in a multi-component reaction-diffusion system. After an overview of past results, we give an exposition of new results on chaotic dynamics of fronts and also a novel type of traveling front solution featuring heterogeneous (instead of constant) tails.

Abstract: According to J. Hadamard’s famous statement, an equation is well-posed if the following are satisfied: i) there exists a solution, ii) the solution is unique, iii) the solution depends continuously on the initial data. In this talk, we carry out these three tasks for BSDEs with jumps. After a brief introduction of BSDEs, in the first part of this talk we will provide existence and uniqueness results for BSDEs with jumps driven by martingales that are stochastically discontinuous, hence we can treat BSDEs and BSΔEs in a unified and general framework. Then, we will present stability results for martingale representations. The third part consists of stability results for solutions of BSDEs not only with respect to the initial data, but also with respect to discretized versions of the driving martingale. In the final part, we will discuss recent results on convergence rates for BSDEs driven by Lévy processes. This talk is based on joint work with Alexandros Saplaouras (Athens), Dylan Possamai (Zürich) and Chenguang Liu (Delft).

Abstract: In the first part, we recall the basics of discrete time Lorentz gases by forcing an analogy with random walks on Z^d. The notion of mixing for such discrete models will be explained in analogy with renewal sequences for random walks on Z^d. We will recall previous results on mixing for finite and infinite horizon Lorentz gases. In the second part, we state and explain the newest result with F. Pene on mixing for the continuous time Lorentz gas with infinite horizon, insisting on the generality of sets allowed in our mixing result.

Abstract: Gaussian processes play an important role in statistics for making inference about spatial or spatiotemporal data. Traditionally, the dependence structures of these random processes (in space or space-​time) are defined via their covariance kernels. Since the computational costs of these kernel-​based approaches for applications such as predictions are, in general, cubic in the number of data points, a vibrant research area has evolved, where various methods for “big data” are proposed. In the last decade, the Stochastic Partial Differential Equation (SPDE) approach has proven to be very efficient for tackling the conflict between limited computing power and desired modeling capabilities. Motivated by a well-​known relation between the Gaussian Matérn class and fractional-​order SPDEs, the key idea of this approach is to define Gaussian processes as solutions to appropriate SPDEs and to use efficient numerical methods, such as the Finite Element Method (FEM) or wavelets, for approximating them. In this talk, I will give an introduction to the (spatial) SPDE approach and discuss several recent developments, in particular with regard to the quality and computational costs of FEM approximations in statistical applications. Finally, I will give an outlook on spatiotemporal models which are based on SPDEs involving fractional powers of parabolic space-​time differential operators. 

This talk is based on joint works with David Bolin, Sonja Cox, Lukas Herrmann, Mihály Kovács, Christoph Schwab and Joshua Willems.

Abstract: In the talk, I will explain how the study of the dynamics of group actions on Cantor sets is motivated by the study of the solutions of systems of PDEs, via the connection with foliation theory.

Abstract: The theory of large deviations is concerned with the limiting behaviour on an exponential scale of a sequence of random variables. In particular, it quantifies exponentially small probabilities of deviations on the scale of the law of large numbers. 

When appropriately scaling a Brownian path, Schilder’s theorem states that this satisfies a large deviation principle. In particular, the exponential rate of decay of the probability of observing a specific path is given by its kinetic energy. 

In this talk, we extend Schilder’s theorem to a Riemannian manifold setting (mainly based on Rik Versendaal. “Large deviations for Brownian motion in evolving Riemannian manifolds”. In: Markov Processes and Relat. Fields 27 (2021), pp. 381–412.). In particular, we allow the Riemannian manifold to evolve over time. To achieve this extension, we first need to define Brownian motion in a (time-evolving) Riemannian manifold. We will mention various ways of doing so, but focus on the ’rolling without slipping’ method, which allows us to transfer a Euclidean Brownian motion onto a Riemannian manifold. Once Riemannian Brownian motion is defined, we will state the extension of Schilder’s theorem to (time-evolving) Riemannian manifolds and give some idea of the proof.