分数噪声驱动的随机热方程解的局部时

数学物理学报 ›› 2019, Vol. 39 ›› Issue (3): 582-595.

分数噪声驱动的随机热方程解的局部时

王志¹(),闫理坦²(),余显烨^3,*()

¹ 宁波工程学院理学院浙江宁波 315211
² 东华大学理学院上海 201620
³ 中山大学数学学院广州 510275

收稿日期:2017-11-02 出版日期:2019-06-26 发布日期:2019-06-27
通讯作者: 余显烨 E-mail:wangzhi1006@hotmail.com;litanyan@hotmail.com;xianyeyu@gmail.com
作者简介:王志, wangzhi1006@hotmail.com|闫理坦, litanyan@hotmail.com
基金资助:
国家自然科学基金(11701304);国家自然科学基金(11571071);国家自然科学基金(11701589);浙江省自然科学基金(LQ16A010006);中央高校基本科研业务费(171gpy17)

Local Times of the Solution to Stochastic Heat Equation with Fractional Noise

Zhi Wang¹(),Litan Yan²(),Xianye Yu^3,*()

¹ School of Sciences, Ningbo University of Technology, Zhejiang Ningbo 315211
² Department of Mathematics, College of Science, Donghua University, Shanghai 201620
³ School of Mathematics, Sun Yat-sen University, Guangzhou 510275

Received:2017-11-02 Online:2019-06-26 Published:2019-06-27
Contact: Xianye Yu E-mail:wangzhi1006@hotmail.com;litanyan@hotmail.com;xianyeyu@gmail.com
Supported by:
the NSFC(11701304);the NSFC(11571071);the NSFC(11701589);the Zhejiang Provincial Natural Science Foundation(LQ16A010006);the Fundamental Research Funds for the Central Universities(171gpy17)

摘要/Abstract

摘要：

该文研究了可加分数噪声驱动的随机热方程解的碰撞局部时和相交局部时.运用局部非确定性和混沌分解等方法得到它们的存在性和光滑性.

关键词: 随机热方程, 分数噪声, 碰撞局部时, 相交局部时, 混沌分解

Abstract:

In this paper, we study the collision and intersection local times of the solution to stochastic heat equation with additive fractional noise. We mainly prove its existence and smoothness properties through local nondeterminism and chaos expansion.

Key words: Stochastic heat equation, Fractional noise, Collision local time, Intersection local time, Chaos expansion

中图分类号:

O211.6

王志,闫理坦,余显烨. 分数噪声驱动的随机热方程解的局部时[J]. 数学物理学报, 2019, 39(3): 582-595.

Zhi Wang,Litan Yan,Xianye Yu. Local Times of the Solution to Stochastic Heat Equation with Fractional Noise[J]. Acta mathematica scientia,Series A, 2019, 39(3): 582-595.

参考文献 25

1	Balan R , Tudor C A . The stochastic heat equation with a fractional-colored noise:existence of the solution. ALEA Lat Am J Probab Math Stat, 2008, 4: 57- 87
2	Berman S M . Local nondeterminism and local times of Gaussian processes. Indiana Univ Math J, 1973, 23: 69- 94 doi: 10.1512/iumj.1974.23.23006
3	Dalang R C . Extending martingale measure stochastic integral with application to spatially homogeneous s.p.d.e.'s. Electron J Probab, 1999, 4: 1- 29 doi: 10.1214/ECP.v4-999
4	Swanson J . Variations of the solution to a stochastic heat equation. Ann Probab, 2007, 35: 2122- 2159 doi: 10.1214/009117907000000196
5	Houdré C , Villa J . An example of infinite dimensional quasi-helix. Comtemp Math, 2003, 336: 195- 202
6	Russo F , Tudor C A . On the bifractional Brownian motion. Stochastic Process Appl, 2006, 5: 830- 856
7	Tudor C A , Xiao Y . Sample path properties of bifractional Brownian motion. Bernoulli, 2007, 13: 1023- 1052 doi: 10.3150/07-BEJ6110
8	Yan L , Gao B , Liu J . The Bouleau-Yor identity for a bi-fractional Brownian motion. Stochastics, 2014, 86: 382- 414 doi: 10.1080/17442508.2013.797424
9	Nualart D , Ortiz-Latorre S . Intersection local time for two indepandent fractional Brownian motions. J Theoret Probab, 2007, 20: 759- 767 doi: 10.1007/s10959-007-0106-x
10	Wu D , Xiao Y . Regularity of intersection local times of fractional Brownian motions. J Theoret Probab, 2010, 23: 972- 1001 doi: 10.1007/s10959-009-0221-y
11	Rosen J . The intersection local time of fractional Brownian motion in the plane. J Multivar Anal, 1987, 23: 37- 46 doi: 10.1016/0047-259X(87)90176-X
12	Hu Y . On the self-intersection local time of fractional Brownian motions-via chaos expansion. J Math Kyoto Univ, 2001, 41: 233- 250 doi: 10.1215/kjm/1250517630
13	Hu Y , Nualart D . Renormalized self-intersection local time for fractional Brownian motions. Ann Probab, 2005, 33: 948- 983 doi: 10.1214/009117905000000017
14	Yan L , Yu X . Derivative for self-intersection local time of multidimensional fractional Brownian motion. Stochastics, 2015, 87: 966- 999 doi: 10.1080/17442508.2015.1019883
15	Jiang Y , Wang Y . Self-intersection local times and collision local times of bifractional Brownian motions. Sci China Ser A, 2009, 52: 1905- 1919 doi: 10.1007/s11425-009-0081-z
16	Shen G , Yan L . Smoothness for the collision local times of bifractional Brownian motions. Sci China Math, 2011, 54: 1859- 1873 doi: 10.1007/s11425-011-4228-3
17	Yan L , Liu J , Chen C . On the collision local time of bifractional Brownian motions. Stoch Dyn, 2009, 9: 479- 491 doi: 10.1142/S0219493709002749
18	Yan L , Shen G . On the collision local time of sub-fractional Brownian Motions. Statist Probab Lett, 2010, 80: 296- 308 doi: 10.1016/j.spl.2009.11.003
19	Chen Z , Wu D , Xiao Y . Smoothness of local times and self-intersection local times of Gaussian random fields. Front Math China, 2015, 10: 777- 805 doi: 10.1007/s11464-015-0487-6
20	Bourguin S , Tudor C A . On the law of the solution to a stochastic heat equation with fractional noise in time. Random Oper Stochastic Equations, 2015, 23: 179- 186
21	Ouahhabi H , Tudor C A . Additive functionals of the solution to fractional stochastic heat equation. J Fourier Anal Appl, 2013, 19: 777- 791 doi: 10.1007/s00041-013-9272-7
22	Tudor C A , Xiao Y . Sample paths of the solution to the fractional-colored stochastic heat equation. Stoch Dyn, 2017, 17: 1- 20
23	Tudor C A . Chaos expansion and regularity of the local time of the solution to the stochastic heat equation with additive fractional-colored noise. Taiwanese J Math, 2013, 17: 1765- 1777 doi: 10.11650/tjm.17.2013.2724
24	Nualart D . Malliavin Calculus and Related Topics. New York: Springer-Verlag, 2006
25	Tudor C A . Analysis of Variations for Self-Similar Processes:A Stochastic Calculus Approach. New York: Springer, 2013

[1]	陈振龙; 李慧琼. 广义α-stable过程自相交局部时增量的Holder律及重点[J]. 数学物理学报, 2006, 26(2): 241-250.
[2]	钟玉泉, 肖益民. Stable单的相交局部时与重点[J]. 数学物理学报, 1995, 15(2): 141-152.