Hölder Continuity for the Parabolic Anderson Model with Space-Time Homogeneous Gaussian Noise

Raluca M. Balan, Lluís Quer-Sardanyons, Jian Song

Research output: Contribution to journalArticleResearch

3 Citations (Scopus)

Abstract

© 2019, Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences. In this article, we consider the Parabolic Anderson Model with constant initial condition, driven by a space-time homogeneous Gaussian noise, with general covariance function in time and spatial spectral measure satisfying Dalang’s condition. First, we prove that the solution (in the Skorohod sense) exists and is continuous in Lp (Ω). Then, we show that the solution has a modification whose sample paths are Hölder continuous in space and time, under the minimal condition on the spatial spectral measure of the noise (which is the same as the condition encountered in the case of the white noise in time). This improves similar results which were obtained in [6, 10] under more restrictive conditions, and with sub-optimal exponents for Hölder continuity.
Original languageEnglish
Pages (from-to)717-730
JournalActa Mathematica Scientia
Volume39
DOIs
Publication statusPublished - 1 May 2019

Keywords

  • 60H07
  • 60H15
  • Gaussian noise
  • Malliavin calculus
  • stochastic partial differential equations

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