Acta mathematica scientia,Series B ›› 2019, Vol. 39 ›› Issue (3): 717-730.doi: 10.1007/s10473-019-0306-3

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HÖLDER CONTINUITY FOR THE PARABOLIC ANDERSON MODEL WITH SPACE-TIME HOMOGENEOUS GAUSSIAN NOISE

Raluca M BALAN1, Lluís QUER-SARDANYONS2, Jian SONG3   

  1. 1. Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, ON, K1N 6N5, Canada;
    2. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra(Barcelona), Catalonia, Spain;
    3. School of Mathematics, Shandong University, Jinan 250100, China
  • Received:2018-07-18 Revised:2019-03-04 Online:2019-06-25 Published:2019-06-27
  • Supported by:
    The first author is supported by a grant from the Natural Sciences and Engineering Research Council of Canada, and the second author is supported by the grant MTM2015-67802P.

Abstract: 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 H¨older continuity.

Key words: Gaussian noise, stochastic partial differential equations, Malliavin calculus

CLC Number: 

  • 60H15
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