Large deviations for symmetric stable processes with Feynman-Kac functionals and its application to pinned polymers

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.