  In Brans-Dicke theory the Universe becomes divided after inflation into many
exponentially large domains with different values of the effective
gravitational constant. Such a process can be described by diffusion equations
for the probability of finding a certain value of the inflaton and dilaton
fields in a physical volume of the Universe. For a typical chaotic inflation
potential, the solutions for the probability distribution never become
stationary but grow forever towards larger values of the fields. We show here
that a non-minimal conformal coupling of the inflaton to the curvature scalar,
as well as radiative corrections to the effective potential, may provide a
dynamical cutoff and generate stationary solutions. We also analyze the
possibility of large nonperturbative jumps of the fluctuating inflaton scalar
field, which was recently revealed in the context of the Einstein theory. We
find that in the Brans--Dicke theory the amplitude of such jumps is strongly
suppressed.
