Small-ball constants, and exceptional flat points of SPDEs

2023/12/10 by Davar Khoshnevisan, Khoshnevisan, Davar, Kunwoo Kim +3 · 5 citations
Economics, Econometrics and Finance · Mathematics · #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2312.05789

Abstract

We study small-ball probabilities for the stochastic heat equation with multiplicative noise in the moderate-deviations regime. We prove the existence of a small-ball constant and related it to other known quantities in the literature. These small-ball estimates are known to imply Chung-type laws of the iterated logarithm (LIL) at typical spatial points; these points can be thought of as "points of flat growth". For this result in a similar context in SPDEs see, for example, the recent work of Chen \citeCh2023. We establish the existence of a new family of exceptional spatial points where the Chung-type LIL fails.

Cited by

• Points of slow growth for parabolic SPDEs
• Chung's LIL for the linear stochastic fractional heat equation at origin
• Reweighting metric measure spaces and Onsager-Machlup
• Local linearization for estimating the diffusion parameter of nonlinear stochastic wave equations with spatially correlated noise
• On the spatio-temporal increments of nonlinear parabolic SPDEs and the open KPZ equation

Related

• Small ball probability estimates for the Hölder semi-norm of the stochastic heat equation
• Small ball probabilities and a support theorem for the stochastic heat equation
• Small Ball Probabilities for the Infinite-Dimensional Ornstein-Uhlenbeck Process in Sobolev Spaces
• Correlation-length bounds, and estimates for intermittent islands in parabolic SPDEs
• Spatial fluctuation for stochastic heat equation with Hölder coefficients