Potential Analysis

, Volume 9, Issue 3, pp 261–291

Asymptotic Properties of Generalized Feynman–Kac Functionals

  • Masayoshi Takeda


Let (\({\mathcal{E}},{\mathcal{F}}\)) be a regular Dirichlet form on L2(X;m) and {Px}x ∈ X the Hunt process generated by (\({\mathcal{E}},{\mathcal{F}}\)). Let μ be a signed 'smooth measure' associated with (\({\mathcal{E}},{\mathcal{F}}\)) and Aμt the continuous additive functional corresponding to the measure μ. Under some conditions on (\({\mathcal{E}},{\mathcal{F}}\)) and μ, we shall prove that

$$\mathop {\lim }\limits_{t \to \infty } \frac{1}{t}\log E_x \left( {\exp \left( { - A_t^\mu } \right)} \right)$$
$$ = - \mathop {\mathop {\inf }\limits_{u \in \mathcal{F}^\mu } }\limits_{\left\| u \right\|_2 = 1} \left( {\mathcal{E}\left( {u,u} \right) + \int_X {\tilde u\tilde u} d\mu } \right)forallx \in X,$$
where \(\mathcal{F}^\mu = \left\{ {u \in \mathcal{F}:\tilde u \in L^2 \left( {X;\left| \mu \right|} \right)} \right\}\)
Generalized Feynman–Kac formula symmetric Markov processes Dirichlet forms. 


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Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Masayoshi Takeda
    • 1
  1. 1.Department of Mathematical Science, Faculty of Engineering ScienceOsaka UniversityJapan

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