The stochastic formalism of inflation allows us to describe the scalar-field dynamics in a non-perturbative way. The correspondence between the diffusion and Schr\"{ o}dinger equations makes it possible to exhaustively construct analytical solutions in stochastic inflation. Those exact statistical quantities such as distribution and correlation functions have one-to-one correspondence to the exactly solvable solutions in non-relativistic quantum mechanics in terms of classical orthogonal polynomials. A class of such solutions is presented by means of isospectral Hamiltonians with an underlying symmetry called shape invariance.
Categories: astro-ph.CO gr-qc hep-th math-ph math.MP nlin.SI