# Probability seminar abstracts

Our seminars are usually held on Mondays at 1.15pm. In normal times they are in room 4W 1.7, but in 2020-21 they will be held via Zoom.

# Amine Asselah

**Large deviation for the intersection of the trace of 2 random walks**

*5th October 2020*

We discuss a Large Deviations Principle for the number of intersections of two independent infinite-time ranges in dimension five and more. This settles, in the discrete setting, a conjecture of van den Berg, Bolthausen and den Hollander. We obtain also some path properties.

(Joint work with B. Schapira.)

# Benjamin Fehrman

**Large-deviations for conservative, stochastic PDE and non-equilibrium fluctuations**

*19th October 2020*

The large deviations of certain interacting particle systems about their hydrodynamic limits have long been formally connected to the small-noise large deviations of certain stochastic PDEs with conservative noise. This relationship had, however, remained non-rigorous due to the fact that such SPDEs are degenerate and supercritical, and therefore lack a suitable solution theory. In this talk, I will explain the well-posedness of a sequence of approximating SPDEs, and I will show that there exists a scaling regime for which the solutions of the approximating equations satisfy a genuine large deviations principle equal to that of the particle process. The results are based on a detailed analysis of the associated skeleton equation - a degenerate parabolic-hyperbolic PDE - in energy critical spaces.

# Laure Dumaz

**Localization of the continuous Anderson hamiltonian in 1-d and its transition towards delocalization**

*2nd November 2020*

We consider the continuous Schrödinger operator - d^{2} / dx^{2} + B’(x) on the interval [0,L] where the potential B’ is a white noise. We study the spectrum of this operator in the large L limit. We show the convergence of the smallest eigenvalues and of the eigenvalues in the bulk towards a Poisson point process, and the localization of the associated eigenvectors in a precise sense. We also find that the transition towards delocalization holds for large eigenvalues of order L, where the limiting law of the point process corresponds to Sch_tau, a process introduced by Kritchevski, Valko and Virag for discrete Schrodinger operators. In this case, the eigenvectors behave like the exponential Brownian motion plus a drift, which proves a conjecture of Rifkind and Virag. Moreover the rescaled and unitarily-changed operator converges (in the strong resolvent sense) towards another differential operator, acting on R^{2} -valued functions and of the form "J\partial_t + 2*2 noise matrix" (where J is the matrix ((0, -1)(1, 0))). Joint works with Cyril Labbé.

# Gonçalo dos Reis

**Large deviations and exit-times for reflected McKean-Vlasov equations with self-stabilizing terms and superlinear drifts**

*16th November 2018*

We study a class of reflected McKean-Vlasov diffusions over a convex domain with self-stabilizing coefficients. This includes coefficients that do not satisfy the classical Wasserstein Lipschitz condition. Further, the process is constrained to a (not necessarily bounded) convex domain by a local time on the boundary. These equations include the subclass of reflected self-stabilizing diffusions that drift towards their mean via convolution of the solution law with a stabilizing potential.
Firstly, we establish existence and uniqueness results for this class and address the propagation of chaos. We work with a broad class of coefficients, including drift terms that are locally Lipschitz in spatial and measure variables. However, we do not rely on the boundedness of the domain or the coefficients to account for these non-linearities and instead use the self-stabilizing properties. We prove a Freidlin-Wentzell type Large Deviations Principle and an Eyring-Kramer's law for the exit-time from subdomains contained in the interior of the reflecting domain.

This is joint work with Daniel Adams, Romain Ravaille, William Salkeld and Julian Tugaut.

# András Tóbiás

**The interplay of dormancy and transfer in bacterial populations: Invasion, fixation and coexistence regimes**

*7th December 2020*

We investigate the interplay between two fundamental mechanisms of microbial population dynamics and evolution called dormancy and horizontal gene transfer. The corresponding traits come in many guises and are ubiquitous in microbial communities, affecting their dynamics in important ways. Recently, they have each moved (separately) into the focus of stochastic individual-based modelling (Billiard et al. 2016, 2018; Champagnat, Méléard and Tran, 2019; Blath and Tóbiás 2019). Here, we examine their combined effects in a unified model. Indeed, we consider the (idealized) scenario of two sub-populations, respectively carrying 'trait 1' and 'trait 2', where trait 1 individuals are able to switch (under competitive pressure) into a dormant state, and trait 2 individuals are able to execute horizontal gene transfer, turning trait 1 individuals into trait 2 ones, at a rate depending on the frequency of individuals.

In the large-population limit, we examine the fate of a single trait i mutant arriving in a trait j resident population living in equilibrium, for i,j=1,2, i \neq j. We provide a complete analysis of the invasion dynamics in all cases where the resident population is individually fit and the initial behaviour of the mutant population is non-critical. We identify parameter regimes for the invasion and fixation of the new trait, stable coexistence of the two traits, and 'founder control' (where the initial resident always dominates, irrespective of its trait). The most striking result is that stable coexistence occurs in certain scenarios even if trait 2 (which benefits from transfer at the cost of trait 1) would be unfit when being merely on its own. In the case of founder control, the limiting dynamical system has an unstable coexistence equilibrium. In all cases, we observe the classical (up to 3) phases of invasion dynamics à la Champagnat 2006. The subject of this talk is joint work with Jochen Blath.

# Ellie Archer

**TBC**

*11th January 2021*