Stochastic approach to de Sitter instability and eternal inflation

Understand

We investigate when effective theories of a scalar field on (quasi-)de Sitter background break down through the stochastic formalism.

• We derive the Fokker-Planck equation leaving the second order time derivative of the scalar field.
• Assuming there exists an equilibrium distribution for the field velocity, we obtain a mean value and a variance of the field velocity caused by the quantum fluctuation.
• Introducing coarse-grained Einstein equations, we obtain bounds for the non-eternal inflation phase and for maintaining the exact de Sitter background.