  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.
