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 $D_{\mathrm{c}}=2$ 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 ${\bar{D}}_{\mathrm{c}}=3/2$ and, in the intermediate values of $D$, ${\bar{D}}_{\mathrm{c}}<D<D_{\mathrm{c}}$, 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.

AMS-LaTeX, 41 pages, 12 figures; This manuscript is prepared for the proceedings of the 9th Oka symposium, held at Nara Women's University, 4-5 December 2010

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