## Tohoku Mathematical Journal 2013 December SECOND SERIES VOL. 65, NO. 4

 Tohoku Math. J. 65 (2013), 467-494

Title LARGE DEVIATIONS FOR SYMMETRIC STABLE PROCESSES WITH FEYNMAN-KAC 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 Green-tight 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 non-negative $\beta$, let $P_{x,t} ^{\beta \mu}$ be the law $X_t$ weighted by the Feynman-Kac 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.