Jordan-Brans-Dicke Stochastic Inflation

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.