Sharp space-time regularity of the solution to stochastic heat equation driven by fractional-colored noise. Issue 4 (3rd July 2020)

• Record Type: Journal Article
• Authors: Herrell, Randall; Song, Renming; Wu, Dongsheng; Xiao, Yimin
• Is Part Of: Stochastic analysis and applications. Volume 38:Issue 4(2020)
• Page Start: 747
• Page End: 768
• Publication Date: 2020-07-03
• Subjects: Stochastic heat equation -- fractional-colored noise -- temporal and spatial regularity -- exact modulus of continuity; 60G15 -- 60J55 -- 60G18 -- 60F25; Stochastic analysis -- Periodicals; 519.2205
• DOI: 10.1080/07362994.2020.1721301
• Languages: English
• ISSNs: 0736-2994

Abstract

Abstract: In this article, we study the following stochastic heat equation ∂ t u ( t, x ) = L u ( t, x ) + B ̇, u ( 0, x ) = 0, 0 ≤ t ≤ T, x ∈ R d, where L is the generator of a Lévy process X in R d, B is a fractional-colored Gaussian noise with Hurst index H ∈ ( 1 2, 1 ) in the time variable and spatial covariance function f which is the Fourier transform of a tempered measure μ . After establishing the existence of solution for the stochastic heat equation, we study the regularity of the solution { u ( t, x ), t ≥ 0, x ∈ R d } in both time and space variables. Under mild conditions, we establish the exact uniform modulus of continuity and a Chung-type law of iterated logarithm for the sample function ( t, x ) ↦ u ( t, x ) . These results, to our knowledge, are new even for the classical stochastic heat equation (where L = Δ ) with space-time white noise and they strengthen the corresponding results of Balan and Tudor (2008) and Tudor and Xiao (2017) where partial regularity results were obtained.