布朗运动可加泛函渐近性的一些新结果

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

设B=(Ω,F,(F_t)_(t≥0),(B_t)_(t≥0),(P_x)_(x∈R^d))为L^2(R^d,m)上经典的布朗运动,(ε,D(ε))为其联系的对称狄氏型.设u∈D(ε),u(B_t)-u(B_0)=M_t^u+N_t^u为u(B_t)的Fukushima分解.该文主要研究由上鞅可乘泛函L_t^(-u):=e~(M_t^(-u)-1/2〈M^(-u〉t)对(B_t)_(t≥0)进行变换所得到的新过程(B_t)_(t≥0)的一些性质;同时还研究了由N_t^u产生的布朗运动可加泛函渐近性问题,并得到了新的结果:如果u有界,▽u∈K_(d-1),且L_t^(-u)是鞅,||E.(e^(M_t^-u))||_q<∞。

Let B=（Ω,F,（F_t）t≥0,（B_t）t≥0,（P_x）x∈R^d）be the classical Brownian motion on L^2（R^d,m）,which is associated with a symmetric Dirichlet form（ε,D（ε））.For u∈D（ε）, u（B_t）-u（-B_0）=M_t^u＋N_t^u is Fukushima decomposition,where u is a quasi-continuous version of u,M_t^u the martingale part and N_t^u the zero energy part.In this paper,the authors first study transformed process B of B,which is determined by the supermartingale L_t^（-u）：=e^M_t^（-u）-1/2（M^（-u）_t, they get some properties of its transition semigroup;Then,they study the asymptotic properties of N_t^u,they get that if L_t^（-u） is a martingale,u is bounded and▽_u∈K_（d-1）,｜E.（e^M_t^（-u））｜｜_q∞, then for every x∈R^d, ■1/t log E_x（e^N_t^u）=-■（ε（f,f）＋ε（f^2,u））, where D（ε）_b=D（ε）∩L~∞（R^d,m）.