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 Bou\`{e}-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.
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.
Sayan Banerjee, A. Budhiraja, Peter Rudzis· 0 citations
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