Keio University

We consider a quantum field model with exponential interactions on the two-dimensional torus,  which is called the ${¥rm{exp}(¥Phi)_{2}$-quantum field model or Hoegh-Krohn’s model. In this talk, we discuss the stochastic quantization of this model. Combining key properties of Gaussian multiplicative chaos with a method for singular SPDEs, we construct a unique time-global solution to the corresponding parabolic stochastic quantization equation in the full $L_{1}$-regime $¥vert ¥alpha ¥vert<{¥sqrt{8¥pi}}$ of the charge parameter $¥alpha$. We also identify the solution with an infinite dimensional diffusion process constructed by the Dirichlet form approach. 

The main part of this talk is based on joint work with Masato Hoshino (Osaka University) and  Seiichiro Kusuoka (Kyoto University), and the full paper can be found on https://link.springer.com/article/10.1007/s00440-022-01126-z

In this talk, we present a graph discretized approximation scheme for diffusions with drift and killing on a complete Riemannian manifold M. More precisely, for a given Schrödinger operator with drift on M having the form A = — Δ — b + V , we introduce a family of discrete time random walks in the  ow generated by the drift b with killing on a sequence of proximity graphs, which are constructed by partitions cutting M into small pieces. As a main result, we prove that the drifted Schrodinger semigroup {e^—tA}_t≥0 is approximated by discrete semigroups generated by the family of random walks with a suitable scale change. This result gives a  nite dimensional summation approximation of a Feynman-Kac type functional integral over M. Furthermore, when M is compact, we also obtain a quantitative error estimate of the convergence.
This talk is based on a joint work with Satoshi Ishiwata (Yamagata University), and the full paper can be found on https://doi.org/10.1007/s00208-024-02809-9.