Affiliation:
University of Tokyo
Date:
Mon, 04/09/2017 - 16:30 - 17:30
Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold. Let $x_0, y_0\in M$ and consider the loop space $P_{x_0,y_0}(M)=\{\gamma\in C([0,1]\to M)~|~\gamma(0)=x_0, \gamma(1)=y_0\}$. Let $\nu^{\lambda}$ be the pinned measure defined by the transition probability $p(t/\lambda,x,y)$, where $p(t,x,y)$ denotes the heat kernel of the diffusion semigroup $e^{t\Delta/2}$. Heuristically, we have
$$ d\nu^{\lambda}_{x_0,y_0}(\gamma)=\frac{1}{Z_{\lambda}} \exp\left(-\lambda E(\gamma)\right) d\gamma, $$