Potential Analysis

, Volume 11, Issue 3, pp 235–247

Exponential Decay of Lifetimes and a Theorem of Kac on Total Occupation Times

  • Masayoshi Takeda


Let \((\tfrac{1}{2}D,H^1 (R^d ))\) 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 (\(A_{\tau _D }^\mu \)).
Exponenial Decay of Lifetime Large Deviation Additive Functional Time Change Dirichlet Form. 


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Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Masayoshi Takeda
    • 1
  1. 1.Department of MathematicsTohoku UniversitySendaiJapan. e-mail

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