TU Berlin

Recent advances at the interface of stochastic analysis, rough path theory, stochastic filtering, stochastic control, and mean-field systems have led to a rapidly developing framework for analyzing stochastic dynamics conditioned on common/observation noise. This mini course  will survey how rough stochastic differential equations, introduced in 2021 by A. Hocquet, K. Lê and the speaker, lead to a unifying perspective across several areas of applied probability. (Additional coauthors include F. Bugini, J. Dause, W. Stannat, H. Zhang and P.Zorin-Kranich).

Recent advances at the interface of stochastic analysis, rough path theory, stochastic filtering, stochastic control, and mean-field systems have led to a rapidly developing framework for analyzing stochastic dynamics conditioned on common/observation noise. This mini course  will survey how rough stochastic differential equations, introduced in 2021 by A. Hocquet, K. Lê and the speaker, lead to a unifying perspective across several areas of applied probability. (Additional coauthors include F. Bugini, J. Dause, W. Stannat, H. Zhang and P.Zorin-Kranich).

Dissipative Hamiltonian systems are an important class of dynamical systems that arise in all areas of science and engineering. They are a special case of port-Hamiltonian control systems. When the system is linearized arround a stationary solution one gets a linear dissipative Hamiltonian typically differential-algebraic system. Despite the fact that the system looks unstructured at first sight, it has remarkable properties.  Stability and passivity are automatic, spectral structures for purely imaginary eigenvalues, eigenvalues at infinity, and even singular blocks in the Kronecker canonical form are very restricted and furthermore the structure leads to fast and efficient iterative solution methods for asociated linear systems. When port-Hamiltonian systems are subject to (structured) perturbations, then it is important to determine the minimal allowed perturbations so that these properties are not preserved. The computation of these structured distances to instability, non-passivity, or non-regularity, is typically a very hard non-convex optimization problem. However, in the context of dissipative Hamiltonian systems, the computation becomes much easier and can even be implemented efficiently for large scale problems in combination with model reduction techniques. We will discuss these distances and the computational methods and illustrate the results via an industrial problem.

 

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"We consider a spatial Lambda-Fleming-Viot process, a model in mathematical biology, with a randomly chosen (rough) selection field. We study the scaling limit of this process in different regimes. This leads to the analysis of semi-discrete approximations of singular SPDEs, in particular the Parabolic Anderson Model and allows to extend previous results to weakly nonlinear cases. The subject presented is based on joint works with Aleksander Klimek and Nicolas Perkowski."

SLE curves are an important family of random curves in the plane. They share many similarites with solutions of SDE (in particular, with Brownian motion). Any quesion asked for the latter can be asked for the former. Inspired by that, Yizheng Yuan and I investigate the support for SLE curves. In this talk, I will explain our theorem with more motivation and idea. 

We show how the theories of paracontrolled distributions and regularity structures can be implemented on manifolds, to solve singular SPDEs like the parabolic Anderson model.

This is ongoing work with Bruce Driver (UCSD) and Antoine Dahlqvist (Cambridge)

Malliavin calculus is implemented in the context of [M. Hairer, A theory of regularity structures, Invent. Math. 2014]. This involves some constructions of independent interest, notably an extension of the structure which accommodates a robust and purely deterministic translation operator in L^2-directions between models. In the concrete context of the generalized parabolic Anderson model in 2D -one of the singular SPDEs discussed in the afore-mentioned article - we establish existence of a density at positive times.

In this talk, I will present recent results that give the necessary mathematical foundation for the study of rough path driven PDEs in the framework of weak solutions. The main tool is a new rough Gronwall Lemma argument whose application is rather wide: among others, it allows to derive the basic energy estimates leading to the proof of existence for e.g. parabolic RPDEs. The talk is based on a joint work with Aurelien Deya, Massimiliano Gubinelli and Samy Tindel.