Related papers: Brownian Motion and Finite Approximations of Quant…
We show that the Schr\"odinger operator associated with a physical system over a local field can be approximated in a very strong sense by finite Schr\"odinger operators. Some striking numerical results are included at the end of the…
We determine the long time behavior and the exact order of the tail probability for the maximal displacement of a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of the associated Schr\"odinger type…
We study pathwise approximation of scalar stochastic differential equations at a single point. We provide the exact rate of convergence of the minimal errors that can be achieved by arbitrary numerical methods that are based (in a…
We prove the convergence of $ \nN $-particle systems of Brownian particles with logarithmic interaction potentials onto a system described by the infinite-dimensional stochastic differential equation (ISDE). % For this proof we present two…
This paper is concerned with Random walk approximations of the Brownian motion on the Affine group Aff(R). We are in particular interested in the case where the innovations are discrete. In this framework, the return probability of the walk…
We describe generalized Brownian motion related to parabolic equation systems from a logical point of view, i.e., as a generalization of Anderson's random walk. The connection to classical spaces is based on the Loeb measure. It seems that…
We obtain the convergence in law of a sequence of excited (also called cookies) random walks toward an excited Brownian motion. This last process is a continuous semi-martingale whose drift is a function, say $\phi$, of its local time. It…
We present a new simple method for rounding a semidefinite programming relaxation of a constraint satisfaction problem. We apply it to the problem of approximate angular synchronization. Specifically, we are given directed distances on a…
We elaborate on the theorem saying that as permeability coefficients of snapping-out Brownian motions tend to infinity in such a way that their ratio remains constant, these processes converge to a skew Brownian motion. In particular,…
We establish the discrete approximation to Brownian motion with varying dimension (BMVD in abbreviation) by random walks. The setting is very similar to that in [11], but here we use a different method allowing us to get rid the…
The indefinite integral of the homogenized Ornstein-Uhlenbeck process is a well-known model for physical Brownian motion, modelling the behaviour of an object subject to random impulses [L. S. Ornstein, G. E. Uhlenbeck: On the theory of…
Let $S_n$ be a lattice random walk with mean zero and finite variance, and let $\Lambda^a_n$ be its occupation measure at level $a$. In this note, we prove local limit theorems for $\Pr[S_n=x,\Lambda^a_n=\ell]$ and…
According to a version of Donsker's theorem, geodesic random walks on Riemannian manifolds converge to the respective Brownian motion. From a computational perspective, however, evaluating geodesics can be quite costly. We therefore…
Approximations of fractional Brownian motion using Poisson processes whose parameter sets have the same dimensions as the approximated processes have been studied in the literature. In this paper, a special approximation to the…
The aim of this paper is to represent any continuous local martingale as an almost sure limit of a nested sequence of simple, symmetric random walks, time changed by a discrete quadratic variation process. One basis of this is a similar…
Since the classical work of L\'evy, it is known that the local time of Brownian motion can be characterized through the limit of level crossings. While subsequent extensions of this characterization have primarily focused on Markovian or…
We provide a rigorous derivation of the brownian motion as the limit of a deterministic system of hard-spheres as the number of particles $N$ goes to infinity and their diameter $\varepsilon$ simultaneously goes to $0$, in the fast…
The analysis of local minima in time series data and random landscapes is essential across numerous scientific disciplines, offering critical insights into system dynamics. Recently, Kundu, Majumdar, and Schehr derived the exact…
After some normalization, the logarithms of the ordered singular values of Brownian motions on $GL(N,\mathbb F)$ with $\mathbb F=\mathbb R, \mathbb C$ form Weyl-group invariant Heckman-Opdam processes on $\mathbb R^N$ of type $A_{N-1}$. We…
In this work we introduce correlated random walks on $\Z$. When picking suitably at random the coefficient of correlation, and taking the average over a large number of walks, we obtain a discrete Gaussian process, whose scaling limit is…