  A determinantal point process (DPP) is an ensemble of random
nonnegative-integer-valued Radon measures, whose correlation functions are all
given by determinants specified by an integral kernel called the correlation
kernel. First we show our new scheme of DPPs in which a notion of partial
isometies between a pair of Hilbert spaces plays an important role. Many
examples of DPPs in one-, two-, and higher-dimensional spaces are demonstrated,
where several types of weak convergence from finite DPPs to infinite DPPs are
given. Dynamical extensions of DPP are realized in one-dimensional systems of
diffusive particles conditioned never to collide with each other. They are
regarded as one-dimensional stochastic log-gases, or the two-dimensional
Coulomb gases confined in one-dimensional spaces. In the second section, we
consider such interacting particle systems in one dimension. We introduce a
notion of determinantal martingale and prove that, if the system has
determinantal martingale representation (DMR), then it is a determinantal
stochastic process (DSP) in the sense that all spatio-temporal correlation
function are expressed by a determinant. In the last section, we construct
processes of Gaussian free fields (GFFs) on simply connected proper subdomains
of $\mathbb{C}$ coupled with interacting particle systems defined on boundaries
of the domains. There we use multiple Schramm--Loewner evolutions (SLEs) driven
by the interacting particle systems. We prove that, if the driving processes
are time-changes of the log-gases studied in the second section, then the
obtained GFF with multiple SLEs are stationary. The stationarity defines an
equivalence relation of GFFs, which will be regarded as a generalization of the
imaginary surface studied by Miller and Sheffield.
