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HOME > Table of Contents and Abstracts > Vol. 65, No. 4
Tohoku Mathematical Journal
2013
December
SECOND SERIES VOL. 65, NO. 4
Tohoku Math. J.
65 (2013), 467494

Title
LARGE DEVIATIONS FOR SYMMETRIC STABLE PROCESSES WITH FEYNMANKAC FUNCTIONALS AND ITS APPLICATION TO PINNED POLYMERS
Author
Yasuhito Nishimori
(Received October 9, 2012, revised January 28, 2013) 
Abstract.
Let $\nu$ and $\mu$ be positive Radon measures on ${\boldsymbol R} ^d$ in Greentight Kato class associated with a symmetric $\alpha$stable process $(X_t , P_x)$ on ${\boldsymbol R}^d$, and $A_t ^\nu$ and $A_t ^\mu$ the positive continuous additive functionals under the Revuz correspondence to $\nu$ and $\mu$. For a nonnegative $\beta$, let $P_{x,t} ^{\beta \mu}$ be the law $X_t$ weighted by the FeynmanKac functional $\exp(\beta A_t ^\mu)$, i.e., $P_{x,t} ^\mu =(Z_{x,t} ^\mu)^{1}\exp(\beta A_t ^\mu)P_x$, where $Z_{x,t} ^\mu$ is a normalizing constant. We show that $A_t ^\nu /t$ obeys the large deviation principle under $P_{x,t}^{\beta \mu}$. We apply it to a polymer model to identify the critical value $\beta _{\rm cr}$ such that the polymer is pinned under the law $P^{\beta \mu} _{x,t} $ if and only if $\beta$ is greater than $\beta_{\rm cr}$. The value $\beta _{\rm cr} $ is characterized by the rate function.
Mathematics Subject Classification.
Primary 60F10; Secondary 82D60, 60G52.
Key words and phrases.
Pinned polymer, large deviations, Dirichlet form, symmetric stable process, additive functional.


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