ON GENERALIZED FEYNMAN-KAC TRANSFORMATION FOR MARKOV PROCESSES ASSOCIATED WITH SEMI-DIRICHLET FORMS

doi:10.1016/S0252-9602(16)30099-6

数学物理学报(英文版) ›› 2016, Vol. 36 ›› Issue (6): 1683-1698.doi: 10.1016/S0252-9602(16)30099-6

ON GENERALIZED FEYNMAN-KAC TRANSFORMATION FOR MARKOV PROCESSES ASSOCIATED WITH SEMI-DIRICHLET FORMS

韩新方, 马丽

Department of Mathematics and Statistics, Hainan Normal University, Haikou 571158, China

收稿日期:2015-06-02 修回日期:2015-10-02 出版日期:2016-12-25 发布日期:2016-12-25
通讯作者: Li MA,E-mail:malihnsd@163.com E-mail:xfanghan@163.com
作者简介:Xinfang HAN,E-mail:xfanghan@163.com
基金资助:
This paper is supported by NSFC (11201102, 11326169, 11361021) and Natural Science Foundation of Hainan Province (112002, 113007).

ON GENERALIZED FEYNMAN-KAC TRANSFORMATION FOR MARKOV PROCESSES ASSOCIATED WITH SEMI-DIRICHLET FORMS

Xinfang HAN, Li MA

Department of Mathematics and Statistics, Hainan Normal University, Haikou 571158, China

Received:2015-06-02 Revised:2015-10-02 Online:2016-12-25 Published:2016-12-25
Contact: Li MA,E-mail:malihnsd@163.com E-mail:xfanghan@163.com
Supported by:
This paper is supported by NSFC (11201102, 11326169, 11361021) and Natural Science Foundation of Hainan Province (112002, 113007).

摘要/Abstract

摘要：

Suppose that X is a right process which is associated with a semi-Dirichlet form (ε, D(ε)) on L²(E; m). Let J be the jumping measure of (ε, D(ε)) satisfying J(E×E-d)<∞. Let u∈D(ε)_b:=D(ε)∩L^∞(E; m), we have the following Fukushima's decomposition ?(X_t)-?(X₀)=M_t^u+N_t^u. Define P_t^uf(x)=E_x[e^{N_t^u}f(X_t)]. Let Q^u(f, g)=ε(f, g)+ε(u, fg) for f, g∈D(ε)_b. In the first part, under some assumptions we show that (Q^u, D(ε)_b) is lower semi-bounded if and only if there exists a constant α₀≥0 such that ||P_t^u||₂≤eα₀t for every t>0. If one of these assertions holds, then (P_t^u)_t≥0 is strongly continuous on L²(E; m). If X is equipped with a differential structure, then under some other assumptions, these conclusions remain valid without assuming J(E×E-d)<∞. Some examples are also given in this part. Let A_t be a local continuous additive functional with zero quadratic variation. In the second part, we get the representation of A_t and give two sufficient conditions for P_t^Af(x)=E_x[e^A_tf(X_t)] to be strongly continuous.

关键词: semi-Dirichlet form, generalized Feynman-Kac semigroup, strong continuity, lower semi-bounded, representation of local continuous additive functional with zero quadratic variation

Abstract:

Key words: semi-Dirichlet form, generalized Feynman-Kac semigroup, strong continuity, lower semi-bounded, representation of local continuous additive functional with zero quadratic variation

中图分类号:

60J55

韩新方, 马丽. ON GENERALIZED FEYNMAN-KAC TRANSFORMATION FOR MARKOV PROCESSES ASSOCIATED WITH SEMI-DIRICHLET FORMS[J]. 数学物理学报(英文版), 2016, 36(6): 1683-1698.

Xinfang HAN, Li MA. ON GENERALIZED FEYNMAN-KAC TRANSFORMATION FOR MARKOV PROCESSES ASSOCIATED WITH SEMI-DIRICHLET FORMS[J]. Acta mathematica scientia,Series B, 2016, 36(6): 1683-1698.

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ON GENERALIZED FEYNMAN-KAC TRANSFORMATION FOR MARKOV PROCESSES ASSOCIATED WITH SEMI-DIRICHLET FORMS

ON GENERALIZED FEYNMAN-KAC TRANSFORMATION FOR MARKOV PROCESSES ASSOCIATED WITH SEMI-DIRICHLET FORMS

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