  The Thoma cone is an infinite-dimensional locally compact space, which is
closely related to the space of extremal characters of the infinite symmetric
group. In another context, the Thoma cone appears as the set of parameters for
totally positive, upper triangular Toeplitz matrices of infinite size.
  The purpose of the paper is to construct a family of continuous time Markov
processes on the Thoma cone, depending on two continuous parameters. Our
construction largely exploits specific properties of the Thoma cone related to
its representation-theoretic origin, although we do not use representations
directly. On the other hand, we were inspired by analogies with random matrix
theory coming from models of Markov dynamics related to orthogonal polynomial
ensembles.
  We show that our processes possess a number of nice properties, namely: (1)
every process X is a Feller process; (2) the infinitesimal generator of X, its
spectrum, and the eigenfunctions admit an explicit description; (3) in the
equilibrium regime, the finite-dimensional distributions of X can be
interpreted as (the laws of) infinite-particle systems with determinantal
correlations; (4) the corresponding time-dependent correlation kernel admits an
explicit expression, and its structure is similar to that of time-dependent
correlation kernels appearing in random matrix theory.
