  While moving down the potential on its classical slow roll trajectory, the
inflaton field is subject to quantum jumps, which take it up or down the
potential at random. In "stochastic inflation", the impact of these quantum
jumps is modeled by smoothing out the field over (at least) Hubble-patch sized
domains and treating fluctuations on smaller scales as noise. The inflaton thus
becomes a stochastic process whose values at a given time are calculated using
its probability distribution. We generalize this approach for non-canonic
kinetic terms of Dirac Born Infeld (DBI) type and investigate the resulting
modifications of the field's trajectory. Since models of DBI inflation arise
from string-inspired scenarios in which the scalar field has a geometric
interpretation, we insist that field value restrictions imposed by the model's
string origin must be respected at the quantum level.
