2025/11/07 by Khoshnevisan, Davar, Lee, Cheuk Yin, Pu, Fei +1
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · doi:10.48550/arxiv.2511.04938
openalex publication_date 2025/11/07 · openalex created_date 2025/11/11 · openalex updated_date 2026/07/28
Consider the following p-dimensional system of Itô type stochastic PDEs, [\beginaligned amp;∂t u(t ,x) = ∂2x u(t ,x) + b(u(t ,x)) + σ(u(t ,x)) ξ(t ,x)
amp;\textfor (t ,x)∈(0 ,∞)×\mathbbT, subject to u(0) ≡ u0 on \mathbbT, \endaligned. where \mathbbT denotes a given one-dimensional torus, the initial data u0:\mathbbT→ℝp is assumed to be fixed and non-random and in C1/2(\mathbbT ;ℝp), and ξ denotes a p-dimensional space-time white noise. Under certain regularity conditions on b and σ, it is proved that, if p ≥ 4, then P\\operatornamedimH u(\t\× F) = 2\operatornamedimH F \text∀compact F⊂\mathbbT, tgt;0\=1. If in addition the matrix σ(v) does not depend on v∈ℝp, and is nonsingular, then the above equality holds for all p≥2.