Random walk after the big bang

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Abstract
The dynamics of inflation is that of a relaxation random process. We examine boundary conditions for this process and give a simple proof for the existence of eternal inflation that takes into account the field dependence of the effective cosmological constant and the finite duration of the inflationary phase. Next, natural initial conditions are formulated that lead to a specific interpretation of the wave function in quantum cosmology. We demonstrate that the Hartle-Hawking wave function describes the equilibrium regime for the stochastic process (with the correct quantum-field-theory limit), but only if the cosmological constant is sufficiently large or if it decays sufficiently slowly. We show in which sense inflation is certain even with the Hartle-Hawking wave function, and propose a new framework for the ‘‘tunneling’’ wave function. On the basis of boundary conditions, we argue that the dynamics of the stochastic phase and, hence, the main features of the present Universe, are independent of the physics above the Planck scale.