Sharp space-time regularity of the solution to stochastic heat equation driven by fractional-colored noise

Randall Herrell, Renming Song, Dongsheng Wu, Yimin Xiao

Research output: Contribution to journalArticle

Abstract

In this article, we study the following stochastic heat equation (Formula presented.) where (Formula presented.) is the generator of a Lévy process X in (Formula presented.) B is a fractional-colored Gaussian noise with Hurst index (Formula presented.) in the time variable and spatial covariance function f which is the Fourier transform of a tempered measure (Formula presented.) After establishing the existence of solution for the stochastic heat equation, we study the regularity of the solution (Formula presented.) 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 (Formula presented.) These results, to our knowledge, are new even for the classical stochastic heat equation (where (Formula presented.)) 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.

Original languageEnglish (US)
JournalStochastic Analysis and Applications
DOIs
StateAccepted/In press - Jan 1 2020

Keywords

  • 60F25
  • 60G15
  • 60G18
  • 60J55
  • exact modulus of continuity
  • fractional-colored noise
  • Stochastic heat equation
  • temporal and spatial regularity

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Applied Mathematics

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