Solutions of SPDE's Associated with a Stochastic Flow

被引：2

作者：

Bhar, Suprio ^{[1
]}

Bhaskaran, Rajeev ^{[2
]}

Sarkar, Barun ^{[2
]}

机构：

[1] Indian Inst Technol Kanpur, Dept Math & Stat, Kanpur 208016, Uttar Pradesh, India

[2] Indian Stat Inst, 8th Mile Mysore Rd, Bangalore 560059, Karnataka, India

来源：

POTENTIAL ANALYSIS | 2020年 / 53卷 / 01期

关键词：

E ' valued process; Hermite-Sobolev space; Mild solution; Strong solution; Monotonicity inequality; S ' valued processes locally of compact support; Stochastic flow; Martingale representation; PROBABILISTIC REPRESENTATIONS; SPACE;

D O I：

10.1007/s11118-019-09764-0

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We consider the following stochastic partial differential equation, associated with a stochastic flow {X (t, x)}, fort > 0, x, as in Rajeev and Thangavelu (Potential Anal. 28(2), 139-162, 2008). We show that the strong solutions constructed there are 'locally of compact support'. Using this notion,we define the mild solutions of the above equation and show the equivalence between strong and mild solutions in the multi Hilbertian space 9/. We show uniqueness of solutions in the case when is smooth via the `monotonicity inequality' for (L*, A*), which is a known criterion for uniqueness.

引用

页码：203 / 221

页数：19

共 25 条

[1] [Anonymous], 2008, THEORY STOCH PROCESS
[2] [Anonymous], 1997, Cambridge Studies in Advanced Mathematics
[3] Characterizing Gaussian Flows Arising from Ito's Stochastic Differential Equations
Bhar, Suprio
[J]. POTENTIAL ANALYSIS, 2017, 46 (02) : 261 - 277
[4] Differential operators on Hermite Sobolev spaces
Bhar, Suprio
Rajeev, B.
[J]. PROCEEDINGS OF THE INDIAN ACADEMY OF SCIENCES-MATHEMATICAL SCIENCES, 2015, 125 (01): : 113 - 125
[5] On interacting systems of Hilbert-space-valued diffusions
Bhatt, AG
Kallianpur, G
Karandikar, RL
Xiong, J
[J]. APPLIED MATHEMATICS AND OPTIMIZATION, 1998, 37 (02) : 151 - 188
[6] Da Prato G., 1992, ENCY MATH ITS APPL, V44
[7] A new approach to the martingale representation theorem
Fitzsimmons, P. J.
Rajeev, B.
[J]. STOCHASTICS-AN INTERNATIONAL JOURNAL OF PROBABILITY AND STOCHASTIC REPORTS, 2009, 81 (05): : 467 - 476
[8] THE MONOTONICITY INEQUALITY FOR LINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
Gawarecki, L.
Mandrekar, V.
Rajeev, B.
[J]. INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS, 2009, 12 (04) : 575 - 591
[9] Gawarecki L, 2011, PROBAB APPL SER, P3, DOI 10.1007/978-3-642-16194-0
[10] Solving the KPZ equation
Hairer, Martin
[J]. ANNALS OF MATHEMATICS, 2013, 178 (02) : 559 - 664

← 1 2 3 →