Outline

The summer school will consist of four minicourses, as well as short talks and a poster session for participants. A tentative schedule is available here.

The minicourses are:

• Mariem Abdellatif and Peter Kuchling: Introduction to the Large Deviations Theory

• Bálint Farkas: Asymptotics and ergodic properties of operator semigroups

• Martin Friesen and Barbara Rüdiger: SPDEs with multiple limiting distributions

• Baris Ugurcan: Singular stochastic PDEs and renormalization of Anderson Hamiltonian

We plan for the school to have both in-person and online attendance. In-person attendance will be subject to local CoViD19 regulations.

For additional information, send an email to the school's address: primer(dot)maths(at)unitn(dot)it.

Minicourses

Introduction to the Large Deviations Theory (Mariem Abdellatif and Peter Kuchling)

Large deviations theory is a part of probability theory that deals with the description of rare events where a sum of random variables deviates from its mean by more than a normal amount i.e., beyond what is described by the central limit theorem (CLT). This theory originated with certain refinements of the central limit theorem obtained by many authors, beginning with Khinchin in 1929, finds application in probability theory, statistics, operation research, information theory, statistical physics, financial mathematics and so on.

In this Summer School, there will be 4 lectures on Large deviations theory. The first lecture will be devoted to giving general definitions such as "Large Deviation, Weak Large Deviation, Rate Functions". The second lecture will be devoted to proving Cramér’s theorem, an important result in Large Deviation. In the third and the fourth lectures, we will study Sanov’s theorem as well as the Method of types.

References:

1. A. Dembo and O. Zeitouni: Large deviations techniques and applications (second edition). Springer, 2010.

2. F. den Hollander: Large deviations. AMS Field Institute Monographs, 2000.

Asymptotics and ergodic properties of operator semigroups (Bálint Farkas)

Strongly continuous one-parameter semigroups of bounded, linear operators, termed usually as C₀-semigroups, provide an effective tool for the study of evolution equations, such as PDEs or SPDEs.
In this minicourse we first offer a very quick introduction to the basics of the theory of C₀-semigroups. Then we move onto the study of their asymptotic (or better to say stability) properties, such as uniform, strong or weak convergence to 0 or to a projection. The last part of the lectures covers some basic facts related to mean ergodicity or other ergodic type properties of C₀-semigroups. To minimize the necessary prerequisites in functional analysis we shall be interested mainly in the Hilbert-space theory, whenever this is helpful. If time allows, the extension to the case of Banach spaces will be briefly touched upon.
Reference to relevant literature will be given during the lectures.

SPDEs with multiple limiting distributions (Martin Friesen and Barbara Rüdiger)

Based on recent joint results with Bálint Farkas and Dennis Schroers we present sufficient conditions for SPDEs to have multiple limiting distributions. In dependence of the initial state we study the convergence in the Wasserstein 2 distance.

Singular stochastic PDEs and renormalization of Anderson Hamiltonian (Baris Ugurcan)

In the first part of the course, we give an introduction to the theory of paracontrolled distributions due to Gubinelli-Imkeller-Perkowski and in the second part we talk about renormalization of the Anderson Hamiltonian (based on joint works with M. Gubinelli and I. Zachhuber).

Talks

Regularization by noise and solutions to singular SDEs (Helena Kremp, Freie Universität Berlin)

In this talk, we introduce the phenomenon of regularization by noise for stochastic differential equations (SDEs). Regularization by noise enables to restore well-posedness of ill-posed ordinary differential equations (ODEs) with low regularity coefficients (non-Lipschitz) by adding Brownian noise to the equation. A classical result in that field is the strong existence and uniqueness of solutions to SDEs with bounded, measurable drift and additive Brownian noise, which was proven by Zvonkin (1974) and Veretennikov (1980, extension to the multidimensional case). We will discuss SDEs with Besov drift of negative regularity (generalized functions) and solutions to the associated singular martingale problem. If time permits, we comment on our current research on rough weak solution for singular Lévy SDEs. The talk is based also on joint work with Nicolas Perkowski.

Tuesday, August 31 2021, 16.45 Room A205

Kolmogorov equations on spaces of measures associated to nonlinear filtering processes (Mattia Martini, Università degli Studi di Milano)

In this talk I will introduce a class of backward Kolmogorov equations on spaces of probability ad positive measures. In particular, such partial differential equations of parabolic type are associated to stochastic processes arising in the context of nonlinear filtering and satisfying some stochastic differential equations, namely the Zakai and the Kushner-Stratonovich equations. I will introduce tools like Itô formulas and then present an existence and uniqueness result of classical solutions to these backward Kolmogorov equations. The talk is based on the preprint: arXiv:2107.11865.

Tuesday, August 31 2021, 17.15 Room A205

Gaussian limits for 2d Directed Polymers (Francesca Cottini, Università degli Studi di Milano-Bicocca)

The Directed Polymer in random environment is a statistical mechanics discrete system which takes its name from Chemistry, where a polymer is a chain of smaller units called monomers. This mathematical model assigns a probability law to the different configurations a polymer can have, taking into account its interaction with a "disorder" given by the environment.

In this talk, we will first introduce the model and in particular its partition function. Then, we will show that the partition function admits a Gaussian scaling limit - we will explain what this convergence result means and some key tools in order to prove it.

Tuesday, August 31 2021, 17.45 Room A205

Central limit theorems for a birth-growth model with Poisson arrivals and random growth speed (Riccardo Turin, University of Bern)

We consider Gaussian approximation in a variant of the classical Johnson-Mehl birth-growth model with random growth speed. Seeds appear randomly in R^d at random times and start growing instantaneously in all directions with a random speed. The location, birth time and growth speed of the seeds are given by a Poisson process. Under suitable conditions on the random growth speed and on a weight function, we provide sufficient conditions for a Gaussian convergence of the sum of the weights at the exposed points, which are those seeds in the model that are not covered at the time of their birth. Moreover, using recent results on stabilization regions, we provide non-asymptotic bounds on the distance between the normalized sum of weights and a standard Gaussian random variable in the Wasserstein and Kolmogorov metrics.

Wednesday, September 1 2021, 16.45 Room A205

Affine pure-jump processes on the cone of positive Hilbert-Schmidt operators (Sven Karbach, University of Amsterdam)

We take a look at a class of affine Markov processes on the cone of positive self-adjoint Hilbert-Schmidt operators. Such processes are well-suited to model the instantaneous variance process in infinite dimensional stochastic volatility models that arise e.g. when considering the dynamics of forward rate functions given by a SPDE in the Heath-Jarrow-Morton-Musiela modeling framework. The existence of a flexible sub-class of affine pure-jump processes on this state-space is carried out in the so called generalized Feller setting, an approach that generalizes the classical Feller setting to non-locally compact state spaces.

Wednesday, September 1 2021, 17.15 Room A205