The Keller Segel model for chemotaxis is a two-dimensional system of parabolic or elliptic PDEs.
Motivated by the study of the fully parabolic model using probabilistic methods, we give rise to a non linear SDE of McKean-Vlasov type with a highly non standard and singular interaction. Indeed, the drift of the equation involves all the past of one dimensional time marginal distributions of the process in a singular way. In terms of approximations by particle systems, an interesting and, to the best of our knowledge, new and challenging difficulty arises: at each time each particle interacts with all the past of the other ones by means of a highly singular space-time kernel.

In this talk, we will analyse the above probabilistic interpretation in $d=1$ and $d=2$.

We discuss a realizationwise correspondence between a Brownian  excursion (conditioned to reach height one) and a triple consisting of

(1) the local time profile of the excursion,

(2) an array of independent time-homogeneous Poisson processes on the real line, and

(3) a fair coin tossing sequence,  where (2) and (3) encode the ordering by height respectively the left-right ordering of the subexcursions.

The three components turn out to be independent,  with (1) giving a time change that is responsible for the time-homogeneity of the Poisson processes.

 By the Ray-Knight theorem, (1) is the excursion of a Feller branching diffusion;  thus the metric structure associated with (2), which generates the so-called lookdown space, can be seen as representing the genealogy underlying the Feller branching diffusion. 

Because of the independence of the three components, up to a time change the distribution of this genealogy does not change under a conditioning on the local time profile. This gives also a natural access to genealogies of continuum populations under competition,  whose population size is modeled e.g. by the Fellerbranching diffusion with a logistic drift.

The lecture is based on joint work with Stephan Gufler and Goetz Kersting.

Planar random growth processes occur widely in the physical world. Examples include diffusion-limited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. One approach to mathematically modelling such processes is to represent the randomly growing clusters as compositions of conformal mappings. In 1998, Hastings and Levitov proposed one such family of models, which includes versions of the physical processes described above. An intriguing property of their model is a conjectured phase transition between models that converge to growing disks, and 'turbulent' non-disk like models. In this talk I will describe a natural generalisation of the Hastings-Levitov family in which the location of each successive particle is distributed according to the density of harmonic measure on the cluster boundary, raised to some power. In recent joint work with Norris and Silvestri, we show that when this power lies within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a limit of stability. This phase transition in fluctuations can be interpreted as the beginnings of a macroscopic phase transition from disks to non-disks analogous to that present in the Hastings-Levitov family.
 

I will introduce the concept of forward rank-dependent performance processes, extending the original notion to forward criteria that incorporate probability distortions and, at the same time, accommodate “real-time” incoming market information. A fundamental challenge is how to reconcile the time-consistent nature of forward performance criteria with the time-inconsistency stemming from probability distortions. For this, I will first propose two distinct definitions, one based on the preservation of performance value and the other on the time-consistency of policies and, in turn, establish their equivalence. I will then fully characterize the viable class of probability distortion processes, providing a bifurcation-type result. This will also characterize the candidate optimal wealth process, whose structure motivates the introduction of a new, distorted measure and a related dynamic market. I will, then, build a striking correspondence between the forward rank-dependent criteria in the original market and forward criteria without probability distortions in the auxiliary market. This connection provides a direct construction method for forward rank-dependent criteria with dynamic incoming information. Furthermore, a direct by-product of our work are new results on the so-called dynamic utilities and time-inconsistent problems in the classical (backward) setting. Indeed, it turns out that open questions in the latter setting can be directly addressed by framing the classical problem as a forward one under suitable information rescaling.

Abstract: Gaussian multiplicative chaos (GMC) has attracted a lot of attention in recent years due to its applications in many areas such as Liouville CFT and random matrix theory, but despite its importance not much has been known about its distributional properties. In this talk I shall explain the study of the tail probability of subcritical GMC and establish a precise formula for the leading order asymptotics, resolving a conjecture of Rhodes and Vargas.

The classic Fatou lemma states that the lower limit of expectations is greater or equal than the expectation of the lower limit for a sequence of nonnegative random variables. This talk describes several generalizations of this fact including generalizations to converging sequences of probability measures. The three types of convergence of probability measures are considered in this talk: weak convergence, setwise convergence, and convergence in total variation. The talk also describes the Uniform Fatou Lemma (UFL) for sequences of probabilities converging in total variation. The UFL states the necessary and sufficient conditions for the validity of the stronger inequality than the inequality in Fatou's lemma. We shall also discuss applications of these results to sequential optimization problems with completely and partially observable state spaces. In particular, the UFL is useful for proving weak continuity of transition probabilities for posterior state distributions of stochastic sequences with incomplete state observations known under the name of Partially Observable Markov Decision Processes. These transition probabilities are implicitly defined by Bayes' formula, and general method for proving their continuity properties have not been available for long time. This talk is based on joint papers with Pavlo Kasyanov, Yan Liang, Michael Zgurovsky, and Nina Zadoianchuk.

 Gaussian processes are well studied object in statistics and mathematics. In Machine Learning, we think of Gaussian processes as prior distributions over functions, which map from the index set to the realised path. To make Gaussian processes a practical tool for machine learning, we have developed tools around variational inference that allow for approximate computation in GPs leveraging the same hardware and software stacks that support deep learning. In this talk I'll give an overview of variational inference in GPs, show some successes of the method, and outline some exciting direction of potential future work.

Description:In his seminal work from the 1950s, William Feller classified all one-dimensional diffusions in terms of their ability to access the boundary (Feller's test for explosions) and to enter the interior from the boundary. Feller's technique is restricted to diffusion processes as the corresponding differential generators allow explicit computations and the use of Hille-Yosida theory. In the present article we study exit and entrance from infinity for jump diffusions driven by a stable process.Many results have been proved for jump diffusions, employing a variety of techniques developed after Feller's work but exit and entrance from infinite boundaries has long remained open. We show that the these processes have features not observes in the diffusion setting. We derive necessary and sufficient conditions on σ so that (i) non-exploding solutions exist and (ii) the corresponding transition semigroup extends to an entrance point at `infinity'. Our proofs are based on very recent developments for path transformations of stable processes via the Lamperti-Kiu representation and new Wiener-Hopf factorisations for Lévy processes that lie therein. The arguments draw together original and intricate applications of results using the Riesz-Bogdan--Żak transformation, entrance laws for self-similar Markov processes, perpetual integrals of Lévy processes and fluctuation theory, which have not been used before in the SDE setting, thereby allowing us to employ classical theory such as Hunt-Nagasawa duality and Getoor's characterisation of transience and recurrence.

Backward Stochastic Differential Equations (BSDEs) have been successfully applied  to represent the value of optimal control problems for controlled

stochastic differential equations. Since in the classical framework several restrictions on the scope of applicability of this method remained, in recent times several approaches have been devised to obtain the desired probabilistic representation in more general situations. We will review the so called  randomization method, originally introduced by B. Bouchard in the framework of optimal switching problems, which consists in introducing an auxiliary,`randomized'' problem with the same value as the original one, where the control process is replaced by an exogenous random point process,and optimization is performed over a family of equivalent probability measures. The value of the randomized problem is then represented

by means of a special class of BSDEs with a constraint on one of the unknown processes.This methodology will be applied in the framework of controlled evolution equations (with immediate applications to controlled SPDEs), a case for which very few results are known so far.