Hiroshi Kawabi

On Asymptotics of Banach Space-valued Itô Functionals of Brownian Rough Paths

Abstract: In this paper, we discuss asymptotics for certain Banach spacevalued Itô functionals of Brownian rough paths based on the results of Inahama-Kawabi [10] and Inahama [9]. Our main tool is the Banach spacevalued rough path theory of T. Lyons. As examples, we deal with heat processes on loop spaces and solutions of the stochastic differential equations (SDEs) on M-type 2 Banach spaces.

A SIMPLE PROOF OF LOG-SOBOLEV INEQUALITIES ON A PATH SPACE WITH GIBBS MEASURES

In this paper, we give a simple proof of log-Sobolev inequalities on an infinite volume path space C (ℝ, ℝd) with Gibbs measures. We introduce a parabolic stochastic partial differential equation which is reversible with respect to the Gibbs measures. In the proof, the gradient estimate for the diffusion semigroup which is derived from the stochastic flow plays a central role.

Strong uniqueness of dirichlet operators related to stochastic quantization under exponential interactions in one-dimensional infinite volume

In this survey paper, we discuss strong uniqueness of Dirichlet operators related to stochastic quantization under exponential (and polynomial) interactions in one-dimensional infinite volume based on joint works with Sergio Albeverio and Michael Rockner (Albeverio et al., J Funct Anal 262:602–638, 2012, [4], Kawabi and Rockner, J Funct Anal 242:486–518, 2007, [11]). We also raise an open problem.