arXiv subjects

Amarjit Budhiraja

Publications and source records attributed to Amarjit Budhiraja.

At least 19 recordsLinked to original sources

arXiv2026

Large Deviations for Markovian Graphon Processes and Associated Dynamical Systems on Networks

We consider temporal models of rapidly evolving Markovian networks whose edge-formation and dissolution rates are determined by time-dependent spatial kernels. Equivalently, these may be viewed as Markovian networks with $O(1)$ jump rates observed over long time horizons. In this regime, paths of graphon-valued processes obtained by averaging over suitable moving time windows provide natural state descriptors. Under appropriate conditions on the jump-rate kernels, we establish laws of large numbers and large deviation principles for these window-averaged paths, both in the weak topology and in the cut metric. We also show that, without such local averaging, the rapidly oscillating graphon process does not satisfy a nontrivial path-space LDP. The resulting rate functions admit explicit and tractable representations, distinct from those arising in static random graph models and finite-horizon dynamic graph models. We further analyze the associated variational problems in several examples and apply the graphon LDP to node-valent dynamical systems driven by the evolving network.

math.PR↗

arXiv2026

Large deviations for long-time occupation measures of stochastic evolution equations with small, asymptotically rough noise

We study the long-time, small-noise behavior of a class of dissipative stochastic evolution equations in a separable Hilbert space, driven by a cylindrical Wiener process whose covariance degenerates to a limiting operator in the strong operator topology. A prototypical example is a stochastic reaction-diffusion equation on a bounded domain with spatially homogeneous but spectrally regularized noise that becomes spatially rough in the limit. We establish a large deviation principle for occupation measures as the time horizon becomes large, the noise intensity tends to zero, and the noise becomes increasingly rough. The result covers a broad class of dissipative equations in infinite dimensions, including those driven by asymptotically rough cylindrical noise whose covariance need not be trace class. Extending the finite-dimensional work of Budhiraja and Zoubouloglou, the infinite-dimensional setting introduces substantial new difficulties: the bound arguments require careful handling of the unbounded evolution and inverse covariance operators, and the increasing roughness of the noise must be balanced against its vanishing amplitude through uniform estimates. Proofs combine analytic semigroup techniques, fractional domain space estimates and a careful treatment of stochastic convolutions in weighted spaces. The rate function is given by a simple explicit formula, the average over the measure of the squared Cameron-Martin cost of canceling the deterministic drift at each point. The proof follows the weak convergence approach based on the Bouè-Dupuis variational formula and constructs near-optimal controls by alternating travel phases that steer the process between prescribed target states and hold phases that stabilize it near a target while shaping the occupation measure.

math.PR↗

arXiv2026

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↗

arXiv2026

Flocking under Fast and Large Jumps: Stability, Chaos, and Traveling Waves

We study a model for flocking given by a $n$-particle system under which each particle jumps forward by a random amount, independently sampled from a given distribution $θ$, with rate given by a non-increasing function $w$ of its signed distance from the system center of mass. This model was introduced in Balázs et. al. (2014) and some of its properties were studied for the case when $w$ is bounded. In the current work we are interested in the setting where $w$ is unbounded, and this feature results in a stochastic dynamical system for interacting particles with fast and large jumps for which little is available in the literature. We characterize the large $n$ limit (the so-called `fluid limit') of the empirical measure process associated with the system and prove a propagation of chaos result. Next, for the centered $n$-particle system, by constructing suitable Lyapunov functions, we establish existence and uniqueness of stationary distributions and study their tail properties. In the special case where $w$ is an exponential function and $θ$ is an exponential distribution, by establishing that all stationary solutions of the McKean-Vlasov equation must be the unique fixed point of the equation, we prove a propagation of chaos result at $t=\infty$ and establish convergence of the particle system, starting from stationarity, in the large $n$ limit, to a traveling wave solution of the McKean-Vlasov equation. The proof of this result may be of interest for other interacting particle systems where convexity properties or functional inequalities generally used for establishing such a result are not available. Our work answers several open problems posed in Balázs et. al.

math.PR↗

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 $λ>0$, the product measure with i.i.d. Exp$(λ)$ 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 $p_\varepsilon^{-1}-1=\exp\{σ\varepsilon^{1/4}\}$, 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↗

arXiv2026

Rank Based Routing in Large Server Systems under Extreme Congestion

We study $n$ parallel queues in an extreme heavy-traffic regime: each server works at rate $n$, while jobs arrive to a dispatcher at rate $n^2-(a-b)\sqrt{n}$, with fixed $a>b>0$. Arrivals are routed by a marginal join-the-shortest-queue policy: a small stream of rate $b\sqrt{n}$ joins the current shortest queue, while the remaining stream of rate $n^2-a\sqrt{n}$ is routed uniformly at random. This policy greatly reduces communication cost relative to full JSQ, while improving load balancing and offering a natural mechanism for premium jobs to join shorter queues. Under diffusive scaling, we prove limit theorems for the ranked queue lengths and associated gap process. The limit is an infinite-dimensional reflected Atlas process, with reflection at the origin and rank-based drift acting on the lowest particle. Its dynamics depend only on $b$, the shortest-queue arrival rate, while $a$ enters through the choice of invariant distribution. We prove well-posedness of this reflected infinite Atlas model and characterize a one-parameter family of product-form stationary gap distributions, parametrized by $a$ and $b$. To connect the diffusion limit with the stationary behavior of the queueing system, we introduce a related "system with pauses'' that agrees with the original dynamics at diffusion scale but admits an exact open Jackson network representation. This yields explicit finite-$n$ stationary gap distributions, whose heavy-traffic limits select the corresponding product-form invariant laws of the infinite reflected Atlas process. As consequences, we obtain sharp asymptotics for the lowest-ranked queues, system imbalance, and average queue length, quantifying the tradeoff between communication cost and load-balancing performance relative to random routing and full join-the-shortest-queue policies.

math.PR↗

arXiv2026

Level 2.5 large deviations and uncertainty relations for non-Markov self-interacting dynamics

We address the general problem of formulating the dynamical large deviations of non-Markovian systems in a closed form. Specifically, we consider a broad class of ``self-interacting'' jump processes whose dynamics depends on the past through a functional of a state-dependent empirical observable. Exploiting a natural separation of timescales, we obtain the exact (so-called ``level 2.5'') large deviation joint statistics of the empirical measure over configurations and of the empirical flux of transitions. As an application of this general framework, we derive explicit general bounds on the fluctuations of trajectory observables, generalising to the non-Markovian case both thermodynamic and kinetic uncertainty relations. We illustrate our theory with simple examples, and discuss potential applications of these results.

cond-mat.stat-mech↗

arXiv2026

Long Time Asymptotics for the Stochastic Follow-the-Leader System

We introduce and analyze a class of interacting particle systems on the real line that combine features of the stochastic rat race and (deterministic) follow-the-leader models. The particle system evolves as a continuous-time pure jump process: the leading particle moves independently, at Exponential jump times, with constant jump rate and iid jump sizes distributed according to a law $θ$, while each of the remaining particles jumps forward, at Exponential times, at rate equal to its distance from the particle immediately ahead, with jump sizes drawn uniformly from the corresponding gap. The dynamics thus encode competition for leadership together with distance-dependent stochastic interactions. Our main focus is the associated gap process, representing the vector of inter-particle distances. We establish the existence of a unique stationary distribution for the gap process and prove uniform geometric ergodicity. Further, when the leader's jump sizes follow an Exponential distribution, we identify the stationary law explicitly as a product of independent Exponential laws, and show that the associated mixing time scales between $Θ(n)$ and $O(n(\log n)^2)$ for an $n$-particle system. As an application of the mixing time results we establish a functional limit theorem that characterizes fluctuations of particle states at large time, under a suitable spatial and temporal scaling and large particle limit. Finally, when the leader's jumps have heavy but integrable tails, we show that each gap has at least one additional finite moment under stationarity than that of the leader's jump size distribution. The model offers a tractable setting for exploring ergodicity, explicit invariant laws, and mixing behavior in non-diffusive particle systems.

math.PR↗

arXiv2025

Jump Processes with Self-Interactions: Large Deviation Asymptotics

We consider a pure jump process $\{X_t\}_{t\ge 0}$ with values in a finite state space $S= \{1, \ldots, d\}$ for which the jump rates at time instant $t$ depend on the occupation measure $L_t \doteq t^{-1} \int_0^t δ_{X_s}\,ds$. Such self-interacting chains arise in many contexts within statistical physics and applied probability. Under appropriate conditions, a large deviation principle is established for the pair $(L_t, R_t)$, as $t \to \infty$, where $R_t$ is the empirical flux process associated with the jump process. We show that the rate function takes a simple form that can be viewed as a dynamical generalization of the classical Donsker and Varadhan rate function for the analogous quantities in the setting of Markov processes, in particular, unlike the Markovian case, the rate function is not convex. Since the state process is non-Markovian, different techniques are needed than in the setting of Donsker and Varadhan and our proofs rely on variational representations for functionals of Poisson random measures and stochastic control methods.

math.PR↗

arXiv2025

Large Deviation Asymptotics for the Supermarket Model with Growing Choices

We consider the Markovian supermarket model with growing choices, where jobs arrive at rate $nλ_n$ and each of $n$ parallel servers processes jobs in its queue at rate $1$. Each incoming job joins the shortest among $d_n \in \{1,\dotsc,n\}$ randomly selected queues. Under the assumption $d_n \to \infty$ and $λ_n \to λ\in (0,\infty)$ as $n\to \infty$, a large deviation principle (LDP) for the occupancy process is established in a suitable infinite-dimensional path space, and it is shown that the rate function is invariant with respect to the manner in which $d_n \to \infty$. The LDP gives information on the rate of decay of probabilities of various types of rare events associated with the system. We illustrate this by establishing explicit exponential decay rates for probabilities of large total number of jobs in the system. As a corollary, we also show that probabilities of certain rare events can indeed depend on the rate of $d_n \to \infty$.

math.PR↗

arXiv2025

Many-Server Asymptotics for Join-the-Shortest-Queue: Large Deviations and Rare Events

The Join-the-Shortest-Queue routing policy is studied in an asymptotic regime where the number of processors $n$ scales with the arrival rate. A large deviation principle (LDP) for the occupancy process is established, as $n\to \infty$, in a suitable infinite-dimensional path space. Model features that present technical challenges include, Markovian dynamics with discontinuous statistics, a diminishing rate property of the transition probability rates, and an infinite-dimensional state space. The difficulty is in the proof of the Laplace lower bound which requires establishing the uniqueness of solutions of certain infinite-dimensional systems of controlled ordinary differential equations. The LDP gives information on the rate of decay of probabilities of various types of rare events associated with the system. We illustrate this by establishing explicit exponential decay rates for probabilities of long queues. In particular, denoting by $E_j^n(T)$ the event that there is at least one queue with $j$ or more jobs at some time instant over $[0,T]$, we show that, in the critical case, for large $n$ and $T$, $\mathbb{P}(E^n_j(T)) \approx \exp\left [-\frac{n (j-2)^2}{4T}\right].$

math.PR↗

arXiv2025

On free boundary problems for the Atlas model

For $n\in\mathbb{N}$, let $\{X^n_i\}$ be an infinite collection of Brownian particles on the real line where the leftmost particle $\min_iX^n_i(t)$ is given a drift $n$, and let $μ^n_t=n^{-1}\sum_iδ_{X^n_i(t)}$, $t\ge0$ denote the normalized configuration measure. The case where the initial particle positions follow a Poisson point process on $[0,\infty)$ of intensity $nλ$, $λ>0$ was studied where it was shown that $μ^n_t$ converge, as $n\to\infty$, to a limit characterized by a Stefan problem of melting solid (respectively, freezing supercooled liquid) type when $λ\ge 2$ (respectively, $0<λ<2$). In this paper it is assumed that $μ^n_0\toμ_0$ in probability, where $μ_0$ is supported on $[0,\infty)$ and satisfies a polynomial growth condition. Because $(y-x)^{-1}μ_0((x,y])$, $0 0$, the free boundary exists as a continuous trajectory, and the process determined by the leftmost particle converges to it.

math.PR↗

arXiv2025

Large Deviations for Empirical Measures of Self-Interacting Markov Chains

Let $Δ^o$ be a finite set and, for each probability measure $m$ on $Δ^o$, let $G(m)$ be a transition probability kernel on $Δ^o$. Fix $x_0 \in Δ^o$ and consider the chain $\{X_n, \; n \in \mathbb{N}_0\}$ of $Δ^o$-valued random variables such that $X_0=x$, and given $X_0, \ldots , X_n$, the conditional distribution of $X_{n+1}$ is $G(L^{n+1})(X_n, \cdot)$, where $L^{n+1} = \frac{1}{n+1} \sum_{i=0}^{n} δ_{X_i}$ is the empirical measure at instant $n$. Under conditions on $G$ we establish a large deviation principle for the empirical measure sequence $\{L^n, \; n \in \mathbb{N}\}$. As one application of this result we obtain large deviation asymptotics for the Aldous-Flannery-Palacios (1988) approximation scheme for quasistationary distributions of irreducible finite state Markov chains. The conditions on $G$ cover various other models of reinforced stochastic evolutions as well, including certain vertex reinforced and edge reinforced random walks and a variant of the PageRank algorithm. The particular case where $G(m)$ does not depend on $m$ corresponds to the classical results of Donsker and Varadhan (1975) on large deviations of empirical measures of Markov processes. However, unlike this classical setting, for the general self-interacting models considered here, the rate function takes a very different form; it is typically non-convex and is given through a dynamical variational formula with an infinite horizon discounted objective function.

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

Functional Central limit theorems for microscopic and macroscopic functionals of inhomogeneous random graphs

We study inhomogeneous random graphs with a finite type space. For a natural generalization of the model as a dynamic network-valued process, the paper establishes the following results: (a) Functional central limit theorems for the infinite vector of microscopic type-densities and characterizations of the limits as infinite-dimensional conditionally Gaussian processes in a certain Banach space. (b) Functional (joint) central limit theorems for macroscopic observables of the giant component in the supercritical regime including size, surplus and number of vertices of various types in the giant component. As a corollary this provides central limit theorems for the size of the largest connected component, its surplus, and its type vector, for percolation on dense graphs obtained from a finite type Graphon. (c) Central limit theorem for the weight of the minimum spanning tree with random i.i.d. Exponential edge weights on dense graph sequences driven by an underlying finite type graphon.

math.PR↗

arXiv2024

Extremal Invariant Distributions of Infinite Brownian Particle Systems with Rank Dependent Drifts

\noindent Consider an infinite collection of particles on the real line moving according to independent Brownian motions and such that the $i$-th particle from the left gets the drift $g_{i-1}$. The case where $g_0=1$ and $g_{i}=0$ for all $i \in \mathbb{N}$ corresponds to the well studied infinite Atlas model. Under conditions on the drift vector $\boldsymbol{g} = (g_0, g_1, \ldots)'$ it is known that the Markov process corresponding to the gap sequence of the associated ranked particles has a continuum of product form stationary distributions $\{π_a^{\boldsymbol{g}}, a \in S^{\boldsymbol{g}}\}$ where $S^{\boldsymbol{g}}$ is a semi-infinite interval of the real line. In this work we show that all of these stationary distributions are extremal and ergodic. We also prove that any product form stationary distribution of this Markov process that satisfies a mild integrability condition must be $π_a^{\boldsymbol{g}}$ for some $a \in S^{\boldsymbol{g}}$. These results are new even for the infinite Atlas model. The work makes progress on the open problem of characterizing all the invariant distributions of general competing Brownian particle systems interacting through their relative ranks. Proofs rely on synchronous and mirror coupling of Brownian particles and properties of the intersection local times of the various particles in the infinite system.

math.PR↗

arXiv2024

Diffusion limits in the quarter plane and non-semimartingale reflected Brownian motion

We consider a continuous-time random walk in the quarter plane for which the transition intensities are constant on each of the four faces $(0,\infty)^2$, $F_1=\{0\}\times(0,\infty)$, $F_2=(0,\infty)\times\{0\}$ and $\{(0,0)\}$. We show that when rescaled diffusively it converges in law to a Brownian motion with oblique reflection direction $d^{(i)}$ on face $F_i$, $i=1,2$, defined via the Varadhan-Williams submartingale problem. A parameter denoted by $α$ was introduced in \cite{vw}, measuring the extent to which $d^{(i)}$ are inclined toward the origin. In the case of the quarter plane, $α$ takes values in $(-2,2)$, and it is known that the reflected Brownian motion is a semimartingale if and only if $α\in(-2,1)$. Convergence results via both the Skorohod map and the invariance principle for semimartingale reflected Brownian motion are known to hold in various settings in arbitrary dimension. In the case of the quarter plane, the invariance principle was proved for $α\in (-2,1)$ whereas for tools based on the Skorohod map to be applicable it is necessary (but not sufficient) that $α\in [-1,1)$. Another tool that has been used to prove convergence in general dimension is the extended Skorohod map, which in the case of the quarter plane provides convergence for $α=1$. This paper focuses on the range $α\in (1,2)$, where the Skorohod problem and the extended Skorohod problem do not possess a unique solution, the limit process is not a semimartingale, and convergence to reflected Brownian motion has not been shown before. The result has implications on the asymptotic analysis of two Markovian queueing models: The {\it generalized processor sharing model with parallelization slowdown}, and the {\it coupled processor model}.

math.PR↗

arXiv2024

Load Balancing in Parallel Queues and Rank-based Diffusions

Consider a system with $K$ parallel queues in which the server for each queue processes jobs at rate $n$ and the total arrival rate to the system is $nK-\upsilon \sqrt{n}$ where $\upsilon \in (0, \infty)$ and $n$ is large. We study rank-based routing policies in which $O(\sqrt{n})$ of the incoming jobs are routed to servers with probabilities depending on their ranked queue-length and the remaining jobs are routed uniformly at random. A particular case, referred to as the marginal join-the-shortest-queue (MJSQ) policy, is one in which the $O(\sqrt{n})$ jobs are routed using the join-the-shortest-queue (JSQ) policy. Our first result provides a heavy traffic approximation theorem for such queuing systems. It turns out that, unlike the JSQ system, there is no state space collapse in the setting of MJSQ (and for the more general rank-based routing schemes) and one obtains a novel diffusion limit which is the constrained analogue of the well studied Atlas model (and other rank-based diffusions) arising from mathematical finance. Next, we prove an interchange of limits result which shows that the steady state of the queuing system is well approximated by that of the limiting diffusion, given explicitly in terms of product laws of Exponential random variables. Using these results, we compute the time asymptotic total queue-length in the heavy traffic limit for the MJSQ system. We find the striking result that although in going from JSQ to MJSQ the communication cost is reduced by a factor of $\sqrt{n}$, the asymptotic total queue-length increases only by a constant factor which can be made arbitrarily close to one by increasing a MJSQ parameter. When the system is overloaded ($\upsilon<0$) we show that although the $K$-dimensional MJSQ system is unstable, the steady state difference between the maximum and minimum queue-lengths stays bounded in probability (in the heavy traffic parameter $n$).

math.PR↗