Talks

Gerold Alsmeyer Branching in random environment, linear fractional distributions and perpetuities

Linear fractional Galton-Watson branching processes in i.i.d. random environment are, on the quenched level, intimately connected to random difference equations by the evolution of the random parameters of their linear fractional marginals. On the other hand, any random difference equation defines an autoregressive Markov chain (a random affine recursion) which can be positive recurrent, null recurrent and transient and which, being the forward iterations of an iterated function system, has an a.s. convergent counterpart in the positive recurrent case given by the corresponding backward iterations. In this talk, I try to provide some insight at how these aspects of random difference equations and their stationary limits, called perpetuities, enter into the results and the analysis, especially in quenched regime. Although most of the results presented are known, I hope that the offered perspective will be welcomed by some participants.

Frank Aurzada Brownian Motion conditioned to spend limited time below a barrier

We condition a Brownian motion with arbitrary starting point y on spending at most \(1\) time unit below \(0\) and provide an explicit description of the resulting process. In particular, we provide explicit formulas for the distributions of its last zero \(g^y\) and of its occupation time \(\Gamma^y\) below \(0\) as functions of \(y\). This generalizes a result of Benjamini and Berestycki from 2011, which covers the special case \( y=0 \). Additionally, we study the behavior of the distributions of \(g^y\) and \(\Gamma^y\), respectively, for \(y\to\pm\infty\). This is joint work with Dominic T. Schickentanz (Darmstadt).

Anita Behme Exponential functionals of Markov additive processes

We consider exponential functionals driven by a bivariate Markov additive process, i.e. driven by a bivariate Lévy process whose characteristics are governed by a continuous-time Markov chain with countable state space. We discuss necessary and sufficient conditions for convergence of such exponential functionals and present conditions for the existence of moments. Moreover, we prove that these exponential functionals appear as stationary distributions of Markov-modulated Ornstein-Uhlenbeck processes. This talk is based on joint work with Apostolos Sideris (Dresden).

Sara Brofferio Affine recursion, branching process and persistence: fruitful interactions

The Markov chain defined by the affine recursion shares with branching processes (in random environment) several asymptotic properties. These similarities have been profitably used by several authors, that were able to transfer results and ideas from one setting to the other. I will show one way to formalize this connection presenting affine recursion and branching processes as two seminal examples of a larger class of Markov processes: the Asymptotically Linear stochastic dynamical systems. In the second part of the talk, I will explain the role of persistence in the study of Asymptotically Linear stochastic dynamical systems in the critical case, in particular to ensure recurrence and describe the tail of invariant measures.

Leif Döring General path integrals and stable SDEs

In recent years there have been several approaches towards the understanding of path integrals for Markov processes. We will discuss a general theorem characterisation finiteness of such integrals. As an applications we give Engelbert-Schmidt type theorems for stable SDEs with \(\alpha<1\).

William Fitzgerald Fredholm Pfaffians and persistence probabilities

I will describe how probabilistic methods can be used to find asymptotics for Fredholm determinants and Pfaffians including both leading order exponents and constant terms. This has applications to persistence probabilities for coalescing and annihilating Brownian motions, Gaussian power series and Kac polynomials. Other applications include the real eigenvalues of non-Hermitian random matrix ensembles.

Sandro Franceschi Reflected Brownian motion in a cone: study of the transient case. Escape and absorption probability, Green's functions and Martin's boundary.

One of the classic problems in the literature devoted to reflected Brownian motion in a two-dimensional cone is the study of its stationary distribution in the recurrent case. On the other hand, we will focus in this talk on the transient case in order to study the Green's functions of this process and their asymptotics. This will naturally lead us to consider the Martin boundary of the process which allows to determine the harmonic functions satisfying oblique Neumann conditions on the edges. For some models, we will illustrate this by studying the probability of escape of the process along an axis or its probability of absorption at the origin. To establish our results, we use analytical methods historically developed in probability and combinatorics to study random walks in the quadrant. We establish functional equations satisfied by the Laplace transforms of Green's functions and the probabilities of escape or absorption. Thanks to the theory of boundary value problems (of Riemann and Carleman) it is possible to determine explicit formulas for these transforms involving hypergeometric functions. The saddle point method and transfer lemmas enable to calculate the asymptotic and to establish the Martin boundary.

Alexander Lindner On quasi-infinitely divisible distributions

A quasi-infinitely divisible distribution on \(\mathbb{R}^d\) is a distribution whose characteristic function can be written as the quotient of the characteristic functions of two infinitely divisible distributions. Equivalently, a probability distribution is quasi-infinitely divisible if and only if its characteristic function admits a Lévy-Khintchine representation with a 'signed Lévy measure'. We will give some examples of quasi-infinitely divisible distributions and address the fact that their collection is dense in the class of probability distributions on \(\mathbb{R}^d\) with respect to weak convergence if and only if the dimension \(d\) equals \(1\). We also give some applications of quasi- infinitely divisible distributions, like obtaining a Cramér-Wold device for \(\mathbb{Z}^d\)-valued probability distributions. The talk is based on works of/with Berger, Kutlu, Sato, Pan and Khartov.

Bastien Mallein Extremal process of the branching Brownian motion in dimension \(d\)

The branching Brownian motion is a particle system in which every particles evolves independently of the other. Each particle moves according to a Brownian motion in dimension \(d\), and split into to after an exponential time, independently of its displacement. We take interest in the asymptotic behavior of the particles that moved furthest from the origin as time grows. We show that these particles split into groups of particles growing in different directions, sampled according to a random measure \(Z(d\theta)\) on the sphere, with norms centered around the atoms of a Poisson point process with exponential intensity.

Sara Mazzonetto Skewed Brownian diffusions: explicit representation of their transition densities

In this talk we first discuss an explicit representation of the transition density of one-dimensional Brownian dynamics undergoing their motion through semipermeable and semireflecting barriers, called skewed Brownian motion. Eventually we discuss extensions and applications considering for instance threshold Bronwnian diffusions (regime-switch at a barrier called threshold) as well. The talk is partially based on joint works with S. Roelly (Potsdam) and D. Dereudre (Lille).

Aleksandar Mijatović Convex minorants and the fluctuation theory of Lévy processes

In this talk I will describe a novel characterisation of the law of the convex minorant of any Lévy process and apply it to establish the Wiener-Hopf factorisation and other key results in the fluctuation theory of Lévy processes. I will also discuss the key steps in the elementary proof of the aforementioned characterisation. This is joint work with Jorge Gonzalez Cazares.

Da Cam Pham The survival probability of a critical multi-type branching process in i.i.d. environment

We study the asymptotic behavior of the probability of non-extinction of a critical multi-type Galton-Watson process in i.i.d. random environments by using limits theorems for products of positive random matrices. Under suitable assumptions, the survival probability is proportional to \(1/\sqrt{n}\). Joint work with Emile Le Page and Marc Peigné.

Christophe Profeta Extremal values for some critical branching Lévy processes

We consider a branching Markov process in continuous time in which the particles evolve independently as Lévy processes. When the branching mechanism is critical or subcritical, the process will eventually die and we may define its overall maximum, i.e. the maximum location ever reached by a particule. We shall give in this talk asymptotic estimates for the survival function of this maximum when the underlying Lévy process is either spectrally negative or stable.

Mladen Savov Asymptotics of densities of exponential functionals of subordinators

In this talk we are going to present the large asymptotic of densities of exponential functionals of subordinators. The intense study of exponential functionals of Lévy processes has been triggered by the different applications these quantities have both in theoretical and applied studies. Gradually, their large asymptotic has been almost completely understood with the omission of the case when the underlying Lévy processes is a subordinator. This is due to the fact that the asymptotic is non-classical from Tauberian point of view. The method we employ is based on the saddle point method which involves the fine understanding of the Stirling-type asymptotic of a new class special functions that we call Bernstein-Gamma functions. Being a natural extension of the classical Gamma function these functions play a role in the study of some Markov self-similar processes and for these reason they have also appeared in fractional calculus as an extension of Caputo’s derivative. This is joint work with Martin Minchev.

Grégory Schehr Statistics of the maximum and the convex hull of a Brownian motion in confined geometries

We consider a Brownian particle with diffusion coefficient \(D\) in a \(d\)-dimensional ball of radius \(R\) with reflecting boundaries. We study the maximum \(M_x(t)\) of the trajectory of the particle along the \(x\)-direction at time \(t\). In the long time limit, the maximum converges to the radius of the ball \(M_x(t) \to R\) for \(t\to \infty\). We investigate how this limit is approached and obtain an exact analytical expression for the distribution of the fluctuations \(\Delta(t) = [R-M_x(t)]/R\) in the limit of large \(t\) in all dimensions. We find that the distribution of \(\Delta(t)\) exhibits a rich variety of behaviors depending on the dimension \(d\). These results are obtained by establishing a connection between this problem and the narrow escape time problem. We apply our results in \(d=2\) to study the convex hull of the trajectory of the particle in a disk of radius \(R\) with reflecting boundaries. We find that the mean perimeter \(\langle L(t)\rangle\) of the convex hull exhibits a slow convergence towards the perimeter of the circle \(2\pi R\) with a stretched exponential decay \(2\pi R-\langle L(t)\rangle \propto \sqrt{R}(Dt)^{1/4} \,e^{-2\sqrt{2Dt}/R}\). Finally, we generalise our results to other confining geometries, such as the ellipse with reflecting boundaries.

Thomas Simon First passage time accross zero for AR(1) sequences

We consider an auto-regressive sequence of order \(1\) with continuous and symmetric innovations, and its first passage time above zero. We show two remarkable factorizations of the generating functions of this passage time depending on the sign of the drift parameter. One factorization extends a classic result by Sparre Andersen on non atomic symmetric random walks. In the case of uniform innovations, we establish a link between the distribution of the passage time and the Mallows-Riordan polynomials. This is based on a joint paper with Gerold Alsmeyer, Alin Bostan and Kilian Raschel.

Vladislav Vysotskiy Persistence of the AR-1 sequences with Rademacher innovations

We find exact asymptotics of the probability that an AR-1 sequence with Rademacher innovations stay positive for long time. This is a joint work with D. Denisov and V. Wachtel.

Vitali Wachtel Supermartingale approach to random walks in cones

I shall discuss a new construction of positive harmonic functions for random walks killed at leaving a cone. This method allows one to construct a positive harmonic function in Lipschitz cones under minimal moment conditions and to obtain more accurate information about the behaviour of the harmonic function not far from the boundary of the cone.