Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises

Min Niu; Bin Xie

doi:10.3934/dcdsb.2018296

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2019, Volume 24, Issue 7: 2989-3009. Doi: 10.3934/dcdsb.2018296

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Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises

Min Niu^1, and
Bin Xie^2, ,

1.
Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, No. 30 Xueyuan Road, Haidian, Beijing 100083, China
2.
Department of Mathematical Sciences, Faculty of Science, Shinshu University, 3-1-1 Asahi, Matsumoto, Nagano 390-8621, Japan

* Corresponding author: Bin Xie
* Corresponding author: Bin Xie

Received: December 31, 2017

Revised: May 31, 2018

Early access: October 2018

Published: May 2019

The first author is supported in part by NSF of China (No.11571030) and the second author is supported by JSPS KAKENH (No.16K05197).

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Abstract

Two properties of stochastic heat equations driven by impulsive noises, which are also called Lévy space-time white noises, are mainly investigated in this paper. We first study the comparison theorem for two stochastic heat equations driven by same noises under some sufficient condition, which is proved via the application of Itô's formula. In particular, we obtain the non-negativity of solutions with non-negative initial data. Then, we investigate the positive correlation of the solutions as the application of the comparison theorem. We prove that the total masses of two solutions relative to two different stochastic heat equations with same noise become nearly uncorrelated after a long time.

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Mathematics Subject Classification: Primary: 60H15; Secondary: 35R60, 60G51.

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