  We study stochastic inflation in the presence of a dynamical gravitational
constant. We describe the Arnowitt--Deser--Misner formalism for
Jordan--Brans--Dicke theory of gravity with an inflaton field. The inflaton and
dilaton scalar fields can be separated into coarse-grained background fields
and quantum fluctuations. We compute the amplitude of the perturbations
generated by those quantum fluctuations in JBD theory with an arbitrary
potential for the inflaton field. The effect of the quantum fluctuations on the
background fields is equivalent to a Brownian motion of the scalar fields,
which can be described with the use of a Fokker--Planck diffusion equation. The
probability to find a given value of the fields in the comoving frame can be
written as a Gaussian distribution centered on their classical trajectory, with
decreasing dispersion along both field directions. We also calculate the
condition for the Universe to enter a self-regenerating inflationary phase. The
probability distribution in the physical frame, which takes into account the
expansion of the proper volume of the inflationary domains, will be
concentrated at the Planck boundary and will move along it towards large values
of the fields.

Astrophysics;General Relativity and Quantum Cosmology;High Energy Physics - Theory