KPZ equation limit of sticky Brownian motion

Abstract

We consider the motion of a particle under a continuum random environment whose distribution is given by the Howitt-Warren flow. In the moderate deviation regime, we establish that the quenched density of the motion of the particle (after appropriate centering and scaling) converges weakly to the $(1+1)$ dimensional stochastic heat equation driven by multiplicative space-time white noise. Our result confirms physics predictions and computations in [LDT17, BLD20] and is the first rigorous instance of such weak convergence in the moderate deviation regime. Our proof relies on a certain Girsanov transform and works for all Howitt-Warren flows with finite and nonzero characteristic measures. Our results capture universality in the sense that the limiting distribution depends on the flow only via the total mass of the characteristic measure. As a corollary of our results, we prove that the fluctuations of the maximum of an $N$-point sticky Brownian motion are given by the KPZ equation plus an independent Gumbel on timescales of order $(\log N{)}^2.$

arXiv comment65 pages, 2 figures

References in corpus (34)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• A theory of regularity structures
• ANOMALOUS SCALING OF THE PASSIVE SCALAR
• Free-energy distribution of the directed polymer at high temperature
• Normal and Anomalous Scaling of the Fourth-Order Correlation Function of a Randomly Advected Passive Scalar
• Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers
• The one-dimensional KPZ equation and its universality class
• Exact height distributions for the KPZ equation with narrow wedge initial condition
• Flows, coalescence and noise
• KPZ reloaded
• Slow modes in passive advection
• The intermediate disorder regime for directed polymers in dimension $1+1$
• Nonlinear fluctuations of weakly asymmetric interacting particle systems
• Random-walk in Beta-distributed random environment
• A simple construction of the continuum parabolic Anderson model on ${\mathbf{R}}^2$
• An almost sure invariance principle for random walks in a space-time random environment
• KPZ equation limit of higher-spin exclusion processes
• On the scaling limits of planar percolation
• The KPZ Limit of ASEP with Boundary
• Multiplicative stochastic heat equations on the whole space
• Nonclassical stochastic flows and continuous products
• Weakly Asymmetric Non-Simple Exclusion Process and the Kardar-Parisi-Zhang Equation
• The crossover from the Macroscopic Fluctuation Theory to the Kardar-Parisi-Zhang equation controls the large deviations beyond Einstein's diffusion
• The random average process and random walk in a space-time random environment in one dimension
• Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation
• Consistent families of Brownian motions and stochastic flows of kernels
• Kardar-Parisi-Zhang equation and large deviations for random walks in weak random environments
• Sticky behavior of fluid particles in the compressible Kraichnan model
• Large Deviations For Sticky Brownian Motions
• Multidimensional sticky Brownian motions as limits of exclusion processes
• Anomalous Fluctuations of Extremes in Many-Particle Diffusion
• Moderate deviations for diffusion in time dependent random media
• Local KPZ behavior under arbitrary scaling limits
• The Brownian web is a two-dimensional black noise

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