  Tunneling processes in de Sitter spacetime are studied by using the
stochastic approach. We exploit the Martin-Siggia-Rose-Janssen-de Dominicis
(MSRJD) functional integral to obtain the tunneling rate. The applicability
conditions of this method are clarified using the Schwinger-Keldysh formalism.
In the case of a shallow potential barrier, we reproduce the Hawking-Moss (HM)
tunneling rate. Remarkably, in contrast to HM picture, the configuration
derived from the MSRJD functional integral satisfies physically natural
boundary conditions. We also discuss the case of a steep potential barrier and
find an interesting Coleman-de Luccia (CDL) bubble-like configuration. Our
results demonstrate how the bubble nucleation process could be described in the
stochastic approach. Our method turns out to be useful for investigating
various tunneling processes during inflation.
