Probability seminar abstracts
Our seminars are usually held on Mondays at 12.15pm in room 4W 1.7.
On uniqueness and blowup properties for a class of second order SDEs
1st October 2018
As the first step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multiplicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution (X_t,Y_t) of the equations
dX_t = Y_t dt,
dY_t = |X_t|a dB_t,
(X_0,Y_0) = (x_0,y_0).
In particular, we prove that solutions are nonunique if 0 < a < 1 and (x_0,y_0) = (0,0), and unique if 1/2 < a and (x_0,y_0) is non-zero. We also show that blowup in finite time holds if a > 1 and (x_0,y_0) is non-zero.
This is a joint work with A. Gomez, J.J. Lee, C. Mueller and M. Salins.
Approximation of stochastic processes by non expansive flows and coming down from infinity
8th October 2018
We are interested in the approximation of semimartingales in finite dimension by dynamical systems. We give trajectorial estimates uniform with respect to the initial condition for a well chosen distance. This relies on a non-expansivity property of the flow and allows to consider non-Lipschitz vector fields. The fluctuations of the process are controlled using the martingale technics initiated in Berestycki et al and stochastic calculus. In particular, we recover and complement known results on Λ-coalescent and birth and death processes in a unified approach. Our main motivation is the study of interacting multi dimensional processes. Using Poincaré’s compactification technics for flows close to infinity, we will classify the coming down from infinity of Lotka-Volterra diffusions and provide uniform estimates for the scaling limits of competitive birth and death processes.
15th October 2018 We introduce a process we call dispersion, which can be used to separate a group of particles located on a single vertex of finite graph. The particles are assumed to be unlabelled and behave identically, with no symmetry breaking properties. The only action they can perform is to move randomly from their current location to a neighbouring vertex when disturbed.
Formally, dispersion is a synchronous process in which M particles are initially placed at a distinguished origin vertex of a graph G. At each time step, at each vertex v occupied by more than one particle at the beginning of this step, each of the particles at the vertex v moves to a neighbour of v chosen independently and uniformly at random. The dispersion process ends at the first step when no vertex is occupied by more than one particle.
For the complete graph K_n , for any constant δ > 1, with high probability, if M ≤ n/2(1 − δ), then the process finishes in O(logn) steps, whereas if M ≥ n/2(1 + δ), then the process needs e^Ω(n) steps to complete (if ever). The lazy variant of this process exhibits analogous behaviour but at a higher threshold. Thus reducing the amount of movement can allow faster dispersion of more particles.
For symmetric graphs such as paths, trees, grids, and hypercubes of large enough size n in terms of the number of particles M, we give bounds on the time to finish dispersion, and the maximum distance traveled from the origin as a function of M.
Joint work with:
Andrew McDowell, Tomasz Radzik, Nicolás Rivera and Takeharu Shiraga.
A numerical scheme for multidimensional SDEs with distributional drift
22nd October 2018
This talk focuses on a multidimensional SDE where the drift is an element of a fractional Sobolev space with negative order, hence a distribution. This SDE admits a unique weak solution in a suitable sense - this was proven in [Flandoli, Issoglio, Russo (2017)]. The aim here is to construct a numerical scheme to approximate this solution. One of the key problems is that the drift cannot be evaluated pointwise, hence we approximate it with suitable functions using Haar wavelets, and then apply (an extended version of) Euler-Maruyama scheme. We then show that the algorithm converges in law, and in the special 1-dimensional case we also get a rate of convergence (and in fact convergence in L1).
This is based on a joint work with T. De Angelis and M. Germain.
Points of infinite multiplicity of a planar Brownian motion
29th October 2018
Points of infinite multiplicity are particular points which the Brownian motion visits infinitely often. Following a work of Bass, Burdzy and Khoshnevisan, we construct and study a measure carried by these points. Joint work with Yueyun Hu and Zhan Shi.
Hydrodynamic limit for an activated exclusion process
5th November 2018
In this talk we present a microscopic model in the family of conserved lattice gases (CLG). Its stochastic short range interaction exhibits a continuous phase transition to an absorbing state at a critical value of the particle density. We prove that, in the active phase (i.e. for initial profiles smooth enough and uniformly larger than the critical density 1/2), the macroscopic behavior of this microscopic dynamics, under periodic boundary conditions and diffusive time scaling, is ruled by a non-linear PDE belonging to the class of fast diffusion equations. The first step in the proof is to show that the system typically reaches an ergodic component in subdiffusive time.
Joint work with O. Blondel, C. Erignoux and M. Sasada.
The extremal point process of branching Ornstein-Uhlenbeck processes
12th November 2018
I will present a recent work on the asymptotic behaviour of extremal particles in a branching Ornstein-Uhlenbeck process: particles move according to an Ornstein-Uhlenbeck process, solution of
dX_s = -\mu X_s ds + dB_s,
and branch at rate 1. We make \mu=\mu_t depend on the time-horizon t at which we observe the particles positions and we suppose that \mu_t t \to \gamma \in (0,\infty]. We show that, properly centred and normalised, the extremal point process continuously interpolates between the extremal point process of the branching Brownian motion (case \gamma=0) and the extremal point process of independent Gaussian random variables (case \gamma=\infty).
Along the way, we obtain several results on standard branching Brownian motion of intrinsic interest.
In particular, we give a probabilistic representation of the probability that the maximal position has an abnormally high velocity.
Joint work with E. Brunet, A. Cortines and B. Mallein.
Paolo Dai Pra
Rhythmic behavior in complex stochastic dynamics
19th November 2018
Interacting particle systems may exhibit, in the thermodynamic limit, a time-periodic behavior in the evolution of their law. We have made an attempt to understand which factors may produce this phenomenon, in particular those related to the time-symmetry breaking of the dynamics: dissipation, delay, asymmetry in the interaction. We will first review some examples of mean field dynamics for which the thermodynamic limit can be explicitly computed and analyzed. Later we will focus on some partial results concerning the first, to our knowledge, example of rhythmic behavior for a system with local interaction: the Ising model with dissipation.