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Local limit theorems for Brownian additive functionals and penalisation of Brownian paths, IX

Bernard Roynette, Marc Yor (2010)

ESAIM: Probability and Statistics

We obtain a local limit theorem for the laws of a class of Brownian additive functionals and we apply this result to a penalisation problem. We study precisely the case of the additive functional: ( A t - : = 0 t 1 X s < 0 d s , t 0 ) . On the other hand, we describe Feynman-Kac type penalisation results for long Brownian bridges thus completing some similar previous study for standard Brownian motion (see [B. Roynette, P. Vallois and M. Yor, Studia Sci. Math. Hung.43 (2006) 171–246]).

Multiplicative functionals of dual processes

Ronald K. Getoor (1971)

Annales de l'institut Fourier

Let X and X ^ be a pair of dual standard Markov processes. We associate to each exact multiplicative function ( M F ) , M of X a unique exact M F , M ^ of X ^ in a natural manner. Any M F , M is assumed to satisfy 0 M t 1 . The map M M ^ is bijective and multiplicative in the sense that: ( M N ) = M ^ N ^ .This correspondence is studied in some detail and several important examples are discussed.These results are then applied to study additive functionals.

Time-dependent Schrödinger perturbations of transition densities

Krzysztof Bogdan, Wolfhard Hansen, Tomasz Jakubowski (2008)

Studia Mathematica

We construct the fundamental solution of t - Δ y - q ( t , y ) for functions q with a certain integral space-time relative smallness, in particular for those satisfying a relative Kato condition. The resulting transition density is comparable to the Gaussian kernel in finite time, and it is even asymptotically equal to the Gaussian kernel (in small time) under the relative Kato condition. The result is generalized to arbitrary strictly positive and finite time-nonhomogeneous transition densities on measure spaces. We...

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