Dating back to Kolmogorov's construction of probabilistic diffusions from solutions of partial differential equations in the early 30's and the joint development of potential theory and Markov process in the early 50's, Analysis and Probability theory have never ceased their fruitful interactions, feeded by practical problems from physics, engineering, finance and others. The inaugural conference of the semester Perspectives in Analysis and Probability will provide a panorama of this domain of interaction, by gathering top international researchers at the interface of Analysis, Probability theory and Partial Differential Equations.
This meeting aims at highlighting some fruitful interactions between probability theory and EDP theory. We shall see how the non-reversibility allows to accelerate the convergence to equilibrium (hypocoercivity) for kinetic equations and their discrete analogs, or how to choose a random initial condition can help to prove existence of dispersive PDE with a very irregular initial condition, and also what we can rigorously prove asymptotic for Hamiltonian systems subjected to stochastic disturbances.
Piecewise Deterministic Markov Processes are non-diffusive stochastic processes which occur naturally in many different models: communication networks, neuronal function, growth of bacterial populations and reliability of complex systems. The main subjects of study relate to the problems of estimation, simulation and asymptotic behaviors (long time, large populations, multi-scale problems) in different application contexts.
Since the first theoretical results for backward stochastic differential equations in the early 90s, this theory has continued to grow due to its applications in the fields of partial differential equations, stochastic geometry, finance and of stochastic control. The conference will provide an opportunity to take stock of recent developments and to consider new perspectives in this field.
Since the pioneering work of K. Itô on the parallel transport along Brownian paths in the early 1960s, the study of interactions between probabilities and differential geometry has become a very rich branch of mathematics. The stochastic approach often proves powerful in (pseudo)-Riemannian Geometry (index formula of Atiyah-Singer, Greew-Wu conjecture) and back recent progress on the geometry of infinite dimensional manifolds help to understand the structure of paths spaces of stochastic processes. The purpose of this meeting is to provide an update on the latest developments at the interface between probability theory and geometry and to initiate new interactions.
This summer school will be mainly devoted to three courses by Ivan Corwin and Martin Hairer on Khardar-Parisi-Zhang (KPZ) equation and Peter Friz on rough paths. Several others talks will present the most recent advances on these topics.
Guiding principle in the courses will be to present the theory of rough paths and recent work by Martin Hairer who managed to solve rigorously the KPZ equation using this theory among other ones. In conjunction with recent breakthroughs on the exact statistics of the KPZ equation (the topic of Ivan Corwin's mini-course), these results will likely validate a number of predictions in the physics literature in which this equation is often seen as a universal limit. The method developed by Martin Hairer will also allow progresses on other very singular equations.