JOINT HÖLDER CONTINUITY OF PARABOLIC ANDERSON MODEL

doi:10.1007/s10473-019-0309-0

Acta mathematica scientia,Series B ›› 2019, Vol. 39 ›› Issue (3): 764-780.doi: 10.1007/s10473-019-0309-0

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JOINT HÖLDER CONTINUITY OF PARABOLIC ANDERSON MODEL

Yaozhong HU¹, Khoa LÊ²

1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G 2G1, Canada;
2. Department of Mathematics, South Kensington Campus, Imperial College London, London, SW7 2AZ, United Kingdom

Received:2018-03-04 Revised:2019-03-03 Online:2019-06-25 Published:2019-06-27
Supported by:
Y. Hu is supported by an NSERC grant and a startup fund of University of Alberta; K. Lê is supported by Martin Hairer's Leverhulme Trust leadership award.

Abstract

Abstract: We obtain the Holder continuity and joint Holder continuity in space and time for the random field solution to the parabolic Anderson equation (_t-1/2△)u=u◇W in d-dimensional space, where W is a mean zero Gaussian noise with temporal covariance γ₀ and spatial covariance given by a spectral density μ(ξ). We assume that γ₀(t) ≤ c|t|^-α₀ and |μ(ξ)| ≤ c|ξ_i|^-α_i or |μ(ξ)| ≤ c|ξ|^-α, where α_i, i=1,…, d (or α) can take negative value.

Key words: Gaussian process, stochastic heat equation, parabolic Anderson model, multiplicative noise, chaos expansion, hypercontractivity, Hölder continuity, joint Hölder continuity

CLC Number:

60H15

Yaozhong HU, Khoa LÊ. JOINT HÖLDER CONTINUITY OF PARABOLIC ANDERSON MODEL[J].Acta mathematica scientia,Series B, 2019, 39(3): 764-780.

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References

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JOINT HÖLDER CONTINUITY OF PARABOLIC ANDERSON MODEL

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