Events

In this talk, I am going to outline the recent history of the theory of early-warning signs for differential equations with a focus on stochastic dynamics near bifurcations. A particular emphasis will be put on the intertwining of the development of this theory with the simultaneous surge of interest in understanding applied geophysical systems and their tipping points.

In the study of Markov processes duality is an important tool used to prove various types of long-time behavior. There exist two approaches to Markov process duality: the algebraic one and the pathwise one. Using the well-known contact process as an example, this talk introduces the general idea of how to construct a pathwise duality for an interacting particle system. Afterwards, several different approaches how to construct pathwise dualities are presented. This is joint work with Jan M. Swart.

In this seminar, we will discuss the estimation of group action using the non-commutative Fourier transform. We'll explore an interesting relationship between the non-commutative Fourier transform and this estimation problem. Next, we'll optimize our estimation method by leveraging this connection under various conditions, including energy constraints. Finally, we'll apply the obtained results to uncertainty relations related to compact groups. Throughout this discussion, we'll highlight the significance of several special functions in this topic.

Abstract We consider the fundamental problem of Linear Quadratic Gaussian Control on an infinite horizon with costly information acquisition. Specifically, we consider a two-dimensional coupled system, where one of the two states is observable, and the other is not. Additionally, to inference from the observable state, costly information is available via an additional, controlled observation process. Mathematically, the Kalman-Bucy filter is used to Markovianize the problem. Using an ansatz, the problem is then reduced to one of the control-dependent, conditional variance for which we show regularity of the value function. Using this regularity for the reduced problem together with the ansatz to solve the problem by dynamic programming and verification and construct the unique optimal control. We analyze the optimal control, the optimally controlled state and the value function and compare various properties to the literature of problems with costly information acquisition. Further, we show existence and uniqueness of an equilibrium for the controlled, conditional variance, and study sensitivity of the control problem at the equilibrium. At last, we compare the problem to the case of no costly information acquisition and fully observable states. Joint work with Christoph Knochenhauer and Yufei Zhang (Imperial College London).