We give analytic and stochastic description of any Markov process that
behaves like a Brownian motion on edges of a star-like graph up to
visiting the vertex. All  possible behaviors  at the graph vertex are
allowed, including continuous and jump exits, delay, killing, and their
combinations, as described by the most general boundary conditions at
this point. We provide an approximation of more complex processes by
simpler ones, and represent each processes in question as a measurable
functions of several one-dimensional Brownian motions, subordinators,
and one exponentially distributed random variable.