STOCHASTIC HEAT EQUATION WITH FRACTIONAL LAPLACIAN AND FRACTIONAL NOISE: EXISTENCE OF THE SOLUTION AND ANALYSIS OF ITS DENSITY

doi:10.1016/S0252-9602(17)30091-7

数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (6): 1545-1566.doi: 10.1016/S0252-9602(17)30091-7

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STOCHASTIC HEAT EQUATION WITH FRACTIONAL LAPLACIAN AND FRACTIONAL NOISE: EXISTENCE OF THE SOLUTION AND ANALYSIS OF ITS DENSITY

刘俊峰¹, Ciprian A. TUDOR²

1. Department of Statistics, Nanjing Audit University, Nanjing 211815, China;
2. Laboratoire Paul Painlevé, Université de Lille 1, F-59655 Villeneuve d'Ascq, France CIMFAV, Universidad de Valparaiso, Chile

收稿日期:2016-07-26 修回日期:2017-03-01 出版日期:2017-12-25 发布日期:2017-12-25
作者简介:刘俊峰,junfengliu@nau.edu.cn;Ciprian A.TUDOR,tudor@math.univ-lille1.fr
基金资助:
Supported by NNSFC (11401313), NSFJS (BK20161579), CPSF (2014M560368, 2015T80475) and 2014 Qing Lan Project; Supported by MEC Project PAI80160047, Conicyt, Chile.

STOCHASTIC HEAT EQUATION WITH FRACTIONAL LAPLACIAN AND FRACTIONAL NOISE: EXISTENCE OF THE SOLUTION AND ANALYSIS OF ITS DENSITY

Junfeng LIU¹, Ciprian A. TUDOR²

1. Department of Statistics, Nanjing Audit University, Nanjing 211815, China;
2. Laboratoire Paul Painlevé, Université de Lille 1, F-59655 Villeneuve d'Ascq, France CIMFAV, Universidad de Valparaiso, Chile

Received:2016-07-26 Revised:2017-03-01 Online:2017-12-25 Published:2017-12-25
Supported by:
Supported by NNSFC (11401313), NSFJS (BK20161579), CPSF (2014M560368, 2015T80475) and 2014 Qing Lan Project; Supported by MEC Project PAI80160047, Conicyt, Chile.

摘要/Abstract

摘要：

In this paper we study a fractional stochastic heat equation on R^d (d ≥ 1) with additive noise (∂/∂t)u(t, x)=D(α/δ)u(t, x) +b(u(t, x)) +?^H(t, x) where D(α/δ) is a nonlocal fractional differential operator and ?^H is a Gaussian-colored noise. We show the existence and the uniqueness of the mild solution for this equation. In addition, in the case of space dimension d=1, we prove the existence of the density for this solution and we establish lower and upper Gaussian bounds for the density by Malliavin calculus.

关键词: stochastic partial differential equation, fractional Brownian motion, Malliavin calculus, Gaussian density estimates

Abstract:

Key words: stochastic partial differential equation, fractional Brownian motion, Malliavin calculus, Gaussian density estimates

刘俊峰, Ciprian A. TUDOR. STOCHASTIC HEAT EQUATION WITH FRACTIONAL LAPLACIAN AND FRACTIONAL NOISE: EXISTENCE OF THE SOLUTION AND ANALYSIS OF ITS DENSITY[J]. 数学物理学报(英文版), 2017, 37(6): 1545-1566.

Junfeng LIU, Ciprian A. TUDOR. STOCHASTIC HEAT EQUATION WITH FRACTIONAL LAPLACIAN AND FRACTIONAL NOISE: EXISTENCE OF THE SOLUTION AND ANALYSIS OF ITS DENSITY[J]. Acta mathematica scientia,Series B, 2017, 37(6): 1545-1566.

参考文献

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STOCHASTIC HEAT EQUATION WITH FRACTIONAL LAPLACIAN AND FRACTIONAL NOISE: EXISTENCE OF THE SOLUTION AND ANALYSIS OF ITS DENSITY

STOCHASTIC HEAT EQUATION WITH FRACTIONAL LAPLACIAN AND FRACTIONAL NOISE: EXISTENCE OF THE SOLUTION AND ANALYSIS OF ITS DENSITY

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