  In genetic circuits, when the mRNA lifetime is short compared to the cell
cycle, proteins are produced in geometrically-distributed bursts, which greatly
affects the cellular switching dynamics between different metastable phenotypic
states. Motivated by this scenario, we study a general problem of switching or
escape in stochastic populations, where influx of particles occurs in groups or
bursts, sampled from an arbitrary distribution. The fact that the step size of
the influx reaction is a-priori unknown, and in general, may fluctuate in time
with a given correlation time and statistics, introduces an additional
non-demographic step-size noise into the system. Employing the probability
generating function technique in conjunction with Hamiltonian formulation, we
are able to map the problem in the leading order onto solving a stationary
Hamilton-Jacobi equation. We show that bursty influx exponentially decreases
the mean escape time compared to the "usual case" of single-step influx. In
particular, close to bifurcation we find a simple analytical expression for the
mean escape time, which solely depends on the mean and variance of the
burst-size distribution. Our results are demonstrated on several realistic
distributions and compare well with numerical Monte-Carlo simulations.
