  We use analytic estimates and numerical simulations to explore the stochastic
approach to vacuum decay. According to this approach, the time derivative of a
scalar field, which is in a local vacuum state, develops a large fluctuation
and the field \flies over" a potential barrier to another vacuum. The
probability distribution for the initial fluctuation is found quantum
mechanically, while the subsequent nonlinear evolution is determined by
classical dynamics. We find in a variety of cases that the rate of such flyover
transitions has the same parametric form as that of tunneling transitions
calculated using the instanton method, differing only by a numerical factor
O(1) in the exponent. An important exception is an "upward" transition from a
de Sitter vacuum to a higher-energy de Sitter vacuum state. The rate of flyover
transitions in this case is parametrically different and can be many orders of
magnitude higher than tunneling. This result is in conflict with the
conventional picture of quantum de Sitter space as a thermal state. Our
numerical simulations indicate that the dynamics of bubble nucleation in
flyover transitions is rather different from the standard picture. The
difference is especially strong for thin-wall bubbles in flat space, where the
transition region oscillates between true and false vacuum until a true vacuum
shell is formed which expands both inwards and outwards, and for upward de
Sitter transitions, where the inflating new vacuum region is contained inside
of a black hole.
