Bessel process, Schramm-Loewner evolution, and Dyson model

Details Bessel process, Schramm-Loewner evolution, and Dyson model Bessel process, Schramm-Loewner evolution, and Dyson model

2011-03-24

arxiv.org/abs/1103.4728v1

Mathematics

Probability

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

Scientific paper

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.

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