Berlin Probability Colloquium

Ort
TU Berlin, Institut für Mathematik
Strasse des 17. Juni 136, Raum MA 043
10623 BerlinZeit
mittwochs, 17.15 Uhr
Programm
27. April 2022
 Fausto Gozzi (LUISS University Rome)
 On mean field control in infinite dimension
 Abstract: The aim of this talk to report on recent work (with A. Cosso, I. Kharroubi, H. Pham, M. Rosestolato) on the optimal control of path dependent McKeanVlasov equations valued in Hilbert spaces. This seems the first paper in this direction We present some motivating examples and the main results (the dynamic programming principle, the law invariance property of the value function V , the Ito formula and the fact that V is a viscosity solution of the HJB equation). We also briefly discuss the issue of uniqueness of the HJB equation presenting a first result in finite dimension. 25. Mai 2022
 Steffen Dereich (WWU Münster)
 Traces left by random walk in the neighbourhood of a vertex
 Abstract: We analyse the visits of a random walk to the neighbourhood of a fixed root vertex. One typically sees the following behaviour in large graphs. When a random walk visits a fixed root vertex it may have further visits on a time scale of order one. Once it leaves the root “significantly” the return time will be of the order of the number of vertices of the graph. In that case we will call the return to the root vertex a new macroscopic visit. In the limit one can distinguish between macroscopic and microscopic visits and we will provide convergence results for a point process that keeps track of the indicidual macroscopic visits including information on each visit on the natural time scale. A standard tool for the analysis of random graphs is local convergence in the sense of Benjamini and Schramm. We show how this concept has to be adapted in order to derive results in our context. Roughly speaking, our results may be applied whenever the mixing time of the random walk is smaller than the number of steps that we can run an exploration algorithm without finding differences to a limitting graph model, with high probability. We mention that our results may be applied for general sparse random graph models. As an example we consider the random graph with fixed degree sequence. Moreover, we discuss related work, in particular, on vacant set percolation. 08. Juni 2022
 Jodi Dianetti (U Bielefeld)
 Multidimensional singular control and related Skorokhod problem: sufficient conditions for the characterization of optimal controls
 Abstract: We characterize the optimal control for a class of singular stochastic control problems as the unique solution to a related Skorokhod reflection problem. The considered optimization problems concern the minimization of a discounted cost functional over an infinite timehorizon through a process of bounded variation affecting an Itôdiffusion. In a multidimensional setting, we prove that the optimal control acts only when the underlying diffusion attempts to exit the socalled waiting region, and that the direction of this action is prescribed by the derivative of the value function. Our approach is based on the study of a suitable monotonicity property of the derivative of the value function through its interpretation as the value of an optimal stopping game. Such a monotonicity allows to construct nearly optimal policies which reflect the underlying diffusion at the boundary of approximating waiting regions. The limit of this approximation scheme then provides the desired characterization. Our result applies to a relevant class of linearquadratic models, among others. Furthermore, it allows to construct the optimal control in degenerate and non degenerate settings considered in the literature, where this important aspect was only partially addressed. This talk is based on a joint work with Giorgio Ferrari. 15. Juni 2022
 Andreas Eberle (Bonn)
 Coupling approaches for Langevin dynamics and nonlinear stochastic differential equations
 Abstract: Couplings are one of the main techniques to quantify the convergence to equilibrium of Markov processes. In this talk, we discuss couplings for (second order) Langevin dynamics and nonlinear stochastic differential equations in the sense of McKean, as well as for the corresponding mean field particle systems. In all these cases, the adequate coupling process has a sticky behaviour on a lower dimensional subspace of the state space. For nonlinear SDE, the coupling distance process is bounded from above by a onedimensional nonlinear diffusion with a sticky boundary at zero. The phase transition of this dominating process can be analyzed explicitly. 22. Juni 2022
 Guillaume Bavarez (HU Berlin)
 The Virasoro structure of Liouville conformal field theory and applications
 Abstract: Liouville CFT is a theory of random surfaces that was introduced by Polyakov in 1981 as a Feynman path integral over the space of metrics on the surface. Recently, a rigorous formulation of the theory was given by Guillarmou, Kupiainen, Rhodes and Vargas; at its core are a Hilbert space (the L^{2}space of a logcorrelated Gaussian field on the circle) and the spectral resolution of a certain Hamiltonian acting on it (a Schrödinger operator with an exponential potential). In the first part of the talk, I will introduce a family of Markov processes (valued in the space of distributions on the circle) whose generators form a representation of the Virasoro algebra on the Hilbert space. As a special case, the Hamiltonian is recovered as the zero mode of the Virasoro algebra. Among other things, this gives a probabilistic interpretation of a representation theoretical result known as the Sugawara construction. In the second part, I will explain how this point of view allows us to reformulate some features of CFT in a geometric way. In particular, it is instrumental in the study of the conformal blocks of the theory, which are universal analytic functions on Teichmüller space serving as building blocks for correlation functions. Based on joint and ongoing work with Guillarmou, Kupiainen, Rhodes and Vargas. 06. Juli 2022
 Ilya Chevyrev (Edinburgh)  16  17 Uhr
 The signature method in machine learning
 Abstract: In this talk, I will give an overview of the socalled signature method in machine learning. The method developed from ideas in rough path theory that the signature serves as a useful and compact summary of a path’s history. Concretely, one can show that the signature arises naturally in Taylor approximations of solutions to ordinary and stochastic differential equations. These results have motivated over the past decade the use of the signature as a feature of timeordered data in machine learning tasks. Surprisingly, the use of the signature as a feature extends far beyond the setting of differential equations, and has had success in fields such as psychiatric data analysis, finance, and character, image, and gesture recognition. I will focus on the universality properties of the signature map, how it can be combined with kernel methods, and its applications to twosample hypothesis teasting.
 Ludovic Tangpi (Princeton)  17.15  18.15 Uhr
 A probabilistic approach to the convergence of large population games to mean field games
 Abstract: Games involving a large population of interacting agents are common around us. They arise in many applications such as physics, epidemiology, economics, or finance. Unfortunately, such games are highly untractable; analytically, but especially computationally. This motivated the introduction of mean field games. These are infinite population idealizations of Nash equilibrium problems in symmetric, finite population games in the microscopic regime. Mean field games present enormous advantages, and their study has given rise to an important literature over the past decade with striking applications. The rigorous understanding of the convergence problem of large population games to mean field games has seen less activity, especially regarding quantitative estimations of the convergence. This talk will review recent progress on this question, with a fully probabilistic method based on a new form of propagation of chaos. 20. Juli 2022
 Gianmario Tessitore (MilanoBicocca))
 Space regularity of Young evolution equations in Banach spaces
 Abstract: We consider the following nonlinear Young equation
 dy(t) = Ay(t)dt + σ(y(t))dx(t), t ∈ (0, T ], with y(0) = ψ,
 A being an unbounded operator and x is ηHölder continuous with η > 0. In [Gubinelli, Tindel, Lejay 2010] it is shown that if ψ is regular enough the above equation admits a unique mild solution with the same regularity as ψ. We show that if the semigroup generated by A has suitable regularizing properties then y(t) is more regular that y(0) and eventually y(t) ∈ D(A) for any t ∈ (0, T ]. As a consequence we get an integral representation of the mild solution y. Then we prove that we can take an less regular initial datum ψ (eventually not regular at all). The main tool in this second part is the construction of a Young convolution integral ∫ t a S(t − r)f (r)dx(r) when the function f blows up as r goes to a+. Joint with Davide Addona and Luca Lorenzi (Università degli Studi di Parma).