arXiv subjects

Peter Rudzis

Publications and source records attributed to Peter Rudzis.

5 recordsLinked to original sources

arXiv2026

Bi-infinite systems of singularly interacting Brownian particles and the KPZ equation

We study a bi-infinite system of interacting Brownian particles on the real line with singular asymmetric interactions mediated by the collision local times. Particles perform Brownian motions, and when neighboring particles collide, the associated local time is split in proportions $p$ and $q=1-p$. We first develop well-posedness theory for the particle system, proving pathwise uniqueness and strong existence under natural growth assumptions on the initial configuration and local times. We also identify a family of stationary distributions for the infinite-dimensional process of gaps between successive particles: for every $\lambda>0$, the product measure with i.i.d. Exp$(\lambda)$ gaps is invariant. Our main result concerns the equilibrium fluctuations of the associated particle-count (height) function in a weakly asymmetric regime. Taking $p=p_\varepsilon$ so that $p_\varepsilon^{-1}-1=\exp\{\sigma \varepsilon^{1/4}\}$, initializing the interparticle gaps with i.i.d. Exp$(1)$ random variables, and applying a microscopic Hopf-Cole transform to the diffusively rescaled count function, we prove convergence, as $\varepsilon \rightarrow 0$, to the multiplicative stochastic heat equation (SHE) with Brownian exponential initial data. Equivalently, the logarithm of the limit is the Hopf-Cole solution of the KPZ equation with two-sided Brownian initial data. The proof combines localization to finite particle subsystems via chains of collisions, Brownian last-passage percolation estimates, a key local time cancellation property, and a martingale problem for a scale-adapted mollification of the Hopf-Cole field, whose space-time regularity is tuned to match that of the limiting SHE. The resulting fluctuation theorem places these Brownian particle systems with asymmetric singular collision dynamics within the KPZ universality class.

math.PR↗

arXiv2025

Fluctuations of the Atlas model from inhomogeneous stationary profiles

The infinite Atlas model describes the evolution of a countable collection of Brownian particles on the real line, where the lowest particle is given a drift of $γ\in [0,\infty)$. We study equilibrium fluctuations for the Atlas model when the system of particles starts from an inhomogeneous stationary profile with exponentially growing density. We show that the appropriately centered and scaled occupation measure of the particle positions, with suitable translations, viewed as a space-time random field, converges to a limit given by a certain stochastic partial differential equation (SPDE). The initial condition for this equation is given by a Brownian motion, the equation is driven by an additive space-time noise that is white in time and colored in space, and the linear operator governing the evolution is the infinitesimal generator of a geometric Brownian motion. We use this SPDE to also characterize the fluctuations of the ranked particle positions with a suitable centering and scaling. Our results describe the behavior of the particles in the bulk and one finds that the Gaussian process describing the asymptotic fluctuations has the same Hölder regularity as a fractional Brownian motion with Hurst parameter $1/4$. One finds that, unlike the setting of a homogeneous profile (Dembo and Tsai (2017)), the behavior on the lower edge of the particle system is very different from the bulk behavior and in fact the variance of the Gaussian limit diverges to $\infty$ as one approaches the lower edge. Indeed, our results show that, with the gaps between particles given by one of the inhomogeneous stationary distributions, the lowest particle, started from $0$, with a linear in time translation, converges in distribution to an explicit non-Gaussian limit as $t\to \infty$.

math.PR↗

arXiv2025

Strong existence, pathwise uniqueness and chains of collisions in infinite Brownian particle systems

We study strong existence and pathwise uniqueness for a class of infinite-dimensional singular stochastic differential equations (SDE), with state space as the cone $\{x \in \mathbb{R}^{\mathbb{N}}: -\infty < x_1 \leq x_2 \leq \cdots\}$, referred to as an infinite system of competing Brownian particles. A `mass' parameter $p \in [0,1]$ governs the splitting proportions of the singular collision local time between adjacent state coordinates. Solutions in the case $p=1/2$ correspond to the well-studied rank-based diffusions, while the general case arises from scaling limits of interacting particle systems on the lattice with asymmetric interactions and the study of the KPZ equation. Under conditions on the initial configuration, the drift vector, and the growth of the local time terms, we establish pathwise uniqueness and strong existence of solutions to the SDE. A key observation is the connection between pathwise uniqueness and the finiteness of `chains of collisions' between adjacent particles influencing a tagged particle in the system. Ingredients in the proofs include classical comparison and monotonicity arguments for reflected Brownian motions, techniques from Brownian last-passage percolation, large deviation bounds for random matrix eigenvalues, and concentration estimates for extrema of Gaussian processes.

math.PR↗

arXiv2023

Semi-deterministic processes with applications in random billiards

We study the ergodic properties of two classes of random dynamical systems: a type of Markov chain which we call the \textit{alternating random walk} and a certain stochastic billiard system which describes the motion of a free-moving rough disk bouncing between two parallel rough walls. Our main results characterize the types of Markov transition kernels which make each system ergodic -- in the first case, with respect to uniform measure on the state space, and in the second case, with respect to Lambertian measure (a classic measure from geometric optics). In addition, building on results from \cite{rudzis2022}, we give explicit examples of rough microstructures which produce ergodic dynamics in the second system. Both systems have the property that the transition kernel governing the dynamics is singular with respect to uniform measure on the state space. As a result, these systems occupy a kind of mean in the problem space between diffusive processes, where establishing ergodicity is relatively easy, and physically realistic deterministic systems, where questions of ergodicity are far less approachable.

math.DS↗

arXiv2022

Rough Collisions

A rough collision law describes the limiting contact dynamics of a pair of rough rigid bodies, as the scale of the rough features (asperities) on the surface of each body goes to zero. The class of rough collision laws is quite large and includes random elements. Our main results characterize the rough collision laws for a freely moving rough disk and a fixed rough wall in dimension 2. Any collision law which (i) is symmetric with respect to a certain well-known invariant measure from billiards theory, and (ii) conserves the projection of the phase space velocity onto the "rolling velocity" is a rough collision law. We also provide a method for explicitly constructing rough collision laws for a broad range of choices of microstructure on the disk and wall. In our introduction, we review past work in billiards, including characterizations of other rough billiard systems, which our results build upon.

math.DS↗