布朗运动可加泛函渐近性的一些新结果

Abstract

Abstract:

Let $\textbf{B=}(\Omega,{{\cal F}},({{\cal F}}_{t})_{t\geq0},(B_{t})_{t\geq0},(P_{x})_{x\inR^{d}})$ be the classical Brownian motion on $L^{2}(R^{d},m)$, which is associated with a symmetric Dirichlet form $({{\cal E}},{{\cal D}}({{\cal E}}))$. For $u\in{{\cal D}}({{\cal E}})$, $\tilde{u}(B_{t})-\tilde{u}(B_{0})=M^{u}_{t}+N^{u}_{t}$ is Fukushima decomposition, where $\tilde{u}$ is a quasi-continuous version of $u$, $M^{u}_{t}$ the martingale part and $N^{u}_{t}$ the zero energy part. In this paper, the authors first study transformed process $\widehat{B}$ of B, which is determined by the supermartingale $L^{-u}_{t}:=e^{M^{-u}_{t}-\frac{1}{2}\langle M^{-u}\rangle_{t}}$, they get some properties of its transition semigroup; Then, they study the asymptotic properties of $N^{u}_{t}$, they get that if $L^{-u}_{t}$ is a martingale, $u$ is bounded and
$\nabla u\in K_{d-1}$, $||E_{.}(e^{M^{-u}_{t}})||_{q}<\infty$, then for every $x\in R^{d}$, \begin{eqnarray*} \lim_{t\rightarrow \infty}\frac{1}{t} \log E_{x}(e^{N^{u}_{t}}) =-\inf_{{f\in {{\cal D}}({\cal E})_{b}}\atop{\| f \| _{L^{2}(R^{d},m)}=1}}({{\cal E}}(f,f)+{{\cal E}}(f^{2},u)), \end{eqnarray*} where
${{\cal D}}({{\cal E}})_{b}={{\cal D}}({{\cal E}})\cap L^{\infty}(R^{d},m)$.

Key words: Dirichlet form, Fukushima decomposition, Brownian motion, Transition density function Asymptotic property

CLC Number:

31C25

CHEN Chuan-Zhong, HAN Xin-Fang, MA Li. Some New Results about Asymptotic Properties of Additive Functionals of Brownian Motion[J].Acta mathematica scientia,Series A, 2010, 30(6): 1485-1494.

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References

[1] Ma Z M, M R\"{o}ckner. Introduction to the Theory of (Non-symmetric) Dirichlet Forms. Berlin: Springer-Verlag, 1992

[2] M Fukushima, Y Oshima, M Takeda. Dirichlet Forms and Symmetric Markov Processes. New York: Walter de Gruyter Berlin, 1994

[3] Sergo Albeverio, Philippe Blanchard, Ma Z M. Feynman-Kac semigroups in terms of signed smooth measures. International Series of Numerical Mathmatics, 1991, 102

[4] M Takeda. Asymptotic properties of generalized Feynman-Kac functionals. Func Anal, 1998, 9: 261--291

[5] Chen Chuanzhong, Ma Z M, Sun Wei. On Girsanov and generalized Feynman-Kac transformations for symmetric Markov processes. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2007, 10(2): 141--163

[6] M Takeda, Zhang T S. Asymptotic properties of additive functionals of Brownian motion. The Analysis of Probability, 1997, 25(2): 940--952

[7] Zhang T S. Generalized Feynman-Kac semigroups, associated quadratic forms and asymptotic properties. Potential Analysis, 2001, 14: 387--408

[8] Chen Chuanzhong, Sun Wei. Strong continuity of generalized Feynman-Kac semigroups: necessary and sufficient conditions. J Func Anal, 2006, 237: 446--465

[9] M J Sharp. General Theory of Markov Process. San Diego: Academic Press, 1988

[10] Chen Z Q, Zhang T S. Girsanov and Feynman-Kac type transformations for symmetric Markov processes. Ann Inst H Poincar\'{e} Probab Statist, 2002, 38(4): 475--505

[11] F C Klebaner. Introduction to Stochastic Calculus with Applications. London: Imperial College Press, 1998: 205--206

[12] Richard F Bass, Chen Z Q. Brownian motion with singular drift. The Annals of Probability, 2003, 31(2): 791--817

[13] Panki Kim, Song Renming. Two-sided estimates on the density of Brownian motion with singular drift. Illionis J M, 2006, 50(3): 635--688

[14] Chen Z Q, PJ Fitzsimmons, M Takeda, J Ying, Zhang T S. Absolute continuity of symmetric Markov process. The Annals of Probability, 2004, 32A(3): 2067--2098

[15] S Albeverio, Ma Z M. Perturbation of Dirichlet forms-lower semiboundedness, closablility, and form cores. J Funct Anal, 1991, 99: 332--356

[16] Song Renming, Glover J, Rao M. Generalized Schr\"{o}dinger Semigroups, in Seminar on Stochastic Processes, 1992. Boston: Birkhar\"{u}ser, 1993

Some New Results about Asymptotic Properties of Additive Functionals of Brownian Motion

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