  We introduce a stochastic dynamics related to the measures that arise in
harmonic analysis on the infinite-dimensional unitary group. Our dynamics is
obtained as a limit of a sequence of natural Markov chains on Gelfand-Tsetlin
graph. We compute finite-dimensional distributions of the limit Markov process,
the generator and eigenfunctions of the semigroup related to this process. The
limit process can be identified with Doob h-transform of a family of
independent diffusions. Space-time correlation functions of the limit process
have a determinantal form.
