Exactly solvable stochastic spectator

The stochastic formalism of inflation allows us to describe the scalar-field dynamics in a non-perturbative way. The correspondence between the diffusion and Schrödinger 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.