Exponential Decay of Lifetimes and a Theorem of Kac on Total Occupation Times
Exponential Decay of Lifetimes and a Theorem of Kac on Total Occupation Times
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寿命的指数衰减和总占用时间的 Kac 定理
DOI：
10.1023/a:1008649623291
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发表时间：
1999
影响因子：
1.1
通讯作者：
M. Takeda
中科院分区：
文献类型：
--
作者：
M. Takeda
摘要让
$$(\tfrac{1}{2}D，H^1(R^d))$$
是Dirichlet积分，并且
$$(B_t，P_z^W)$$
R上的布朗运动设μ是Kato类中的有限正测度
$$A_t^\MU$$
与μ有关的加法泛函。我们证明了对于RD的正则域D
$$\Begin{Gathered}\Mathop{\LIM}\Limits_{\beta\to\inty}\frac{1}{\beta}\log P_z^W(A_{\tau_D}^\Mu&>；\beta)\；\；=\；-\inf\Left\{{\tfrac{1}{2}D(u，u)：u\in C_0^\inty(D)\int_D{u^2{\Text{d}\Mu=1}\Right\}\hFill\{\Text{for any}}x\in D，\hFill\end{gathered}$$
其中τD是从D的退出时间。作为应用，我们考虑维纳泛函EXP(
$$A_{\tau_D}^\MU$$
)。
AbstractLet
$$(\tfrac{1}{2}D,H^1 (R^d ))$$
be the Dirichlet integral and
$$(B_t ,P_z^W )$$
the Brownian motion on R. Let μ be a finite positive measure in the Kato class and
$$A_t^\mu $$
the additive functional associated with μ. We prove that for a regular domain D of Rd
$$\begin{gathered} \mathop {\lim }\limits_{\beta \to \infty } \frac{1}{\beta }\log P_z^W (A_{\tau _D }^\mu > \beta )\;\; = \;\; - \inf \left\{ {\tfrac{1}{2}D(u,u):u \in C_0^\infty (D)\int_D {u^2 {\text{d}}} \mu = 1} \right\} \hfill \\ {\text{ for any }}x \in D, \hfill \\ \end{gathered} $$
where τD is the exit time from D. As an application, we consider the integrability of Wiener functional exp (
$$A_{\tau _D }^\mu $$
).