Bessel process, Schramm-Loewner evolution, and Dyson model

Abstract

Bessel process is defined as the radial part of the Brownian motion (BM) in the $D$-dimensional space, and is considered as a one-parameter family of one-dimensional diffusion processes indexed by $D$, BES$^{(D)}$. It is well-known that $D_{\rm c}=2$ is the critical dimension. Bessel flow is a notion such that we regard BES$^{(D)}$ with a fixed $D$ as a one-parameter family of initial value. There is another critical dimension $\bar{D}_{\rm c}=3/2$ and, in the intermediate values of $D$, $\bar{D}_{\rm c} < D < D_{\rm c}$, behavior of Bessel flow is highly nontrivial. The dimension D=3 is special, since in addition to the aspect that BES$^{(3)}$ is a radial part of the three-dimensional BM, it has another aspect as a conditional BM to stay positive. Two topics in probability theory and statistical mechanics, the Schramm-Loewner evolution (SLE) and the Dyson model (Dyson's BM model with parameter $\beta=2$), are discussed. The SLE$^{(D)}$ is introduced as a 'complexification' of Bessel flow on the upper-half complex-plane. The Dyson model is introduced as a multivariate extension of BES$^{(3)}$. We explain the 'parenthood' of BES$^{(D)}$ and SLE$^{(D)}$, and that of BES$^{(3)}$ and the Dyson model. It is shown that complex analysis is effectively applied to study stochastic processes and statistical mechanics models in equilibrium and nonequilibrium states.