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The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its…
Let $d\geq 2$. In this paper, we investigate the following stochastic differential equation (SDE) in ${\mathbb R}^d$ driven by Brownian motion $$ {\rm d} X_t=b(t,X_t){\rm d} t+\sqrt{2}{\rm d} W_t, $$ where $b$ belongs to the space ${\mathbb…
We provide sufficient conditions on the coefficients of a stochastic functional differential equation with bounded memory driven by Brownian motion which guarantee existence and uniqueness of a maximal local and global strong solution for…
We consider the eigenvectors of the principal minor of dimension $n< N$ of the Dyson Brownian motion in $\mathbb{R}^{N}$ and investigate their asymptotic overlaps with the eigenvectors of the full matrix in the limit of large dimension. We…
We consider a reflected backward stochastic differential equations with default time and an optional barrier in a filtration generated by a one-dimensional Brownian motion and a defaultable process. We suppose that the barrier have…
Understanding the behavior of stochastic gradient methods is a central problem in modern machine learning. Recent work has highlighted diagonal linear networks as a simplified yet expressive setting for analyzing the optimization and…
We study the scaling limits of stochastic gradient descent (SGD) with constant step-size in the high-dimensional regime. We prove limit theorems for the trajectories of summary statistics (i.e., finite-dimensional functions) of SGD as the…
We obtain Fisher-Hartwig asymptotics with root and jump type singularities in space-time under the law of the stationary Hermitian Ornstein-Uhlenbeck process, which serve as a dynamical generalization of earlier static results obtained by…
In the recent article [Hairer, M., Hutzenthaler, M., Jentzen, A., Loss of regularity for Kolmogorov equations, Ann. Probab. 43 (2015), no. 2, 468--527] it has been shown that there exist stochastic differential equations (SDEs) with…
We study differential equations with a linear, path dependent drift and discrete delay in the diffusion term driven by a $\gamma$-H\"older rough path for $\gamma > \frac{1}{3}$. We prove well-posedness of these systems and establish a…
We construct an almost sure bijection that recovers the directed landscape on the half-plane from a sequence of independent Brownian motions. This map is the natural scaling limit of the Robinson--Schensted--Knuth (RSK) correspondence. The…
We study the scaling limit of a branching random walk in static random environment in dimension $d=1,2$ and show that it is given by a super-Brownian motion in a white noise potential. In dimension $1$ we characterize the limit as the…
We study a symmetrized (half-space) version of geometric last passage percolation with a boundary parameter $c$ that interpolates between subcritical, critical, and supercritical behavior. This model gives rise to a family of interlacing…
We obtain well-posedness results for a class of ODE with a singular drift and additive fractional noise, whose right-hand-side involves some bounded variation terms depending on the solution. Examples of such equations are reflected…
We consider Dyson Brownian motion for classical values of $\beta$ with deterministic initial data $V$. We prove that the local eigenvalue statistics coincide with the GOE/GUE in the fixed energy sense after time $t \gtrsim 1/N$ if the…
We show the convergence of the characteristic polynomial for random permutation matrices sampled from the generalized Ewens distribution. Under this distribution, the measure of a given permutation depends only on its cycle structure,…
In this paper, we introduce a new simple approach to developing and establishing the convergence of splitting methods for a large class of stochastic differential equations (SDEs), including additive, diagonal and scalar noise types. The…
We consider two related linear PDE's perturbed by a fractional Brownian motion. We allow the drift to be discontinuous, in which case the corresponding deterministic equation is ill-posed. However, the noise will be shown to have a…
We consider a stochastic flow driven by a finite dimensional Brownian motion. We show that almost every realization of such a flow exhibits strong statistical properties such as the exponential convergence of an initial measure to the…
In this paper we present a new and flexible method to show that, in one dimension, various self-repellent random walks converge to self-repellent Brownian motion in the limit of weak interaction after appropriate space-time scaling. Our…