Analytical Solution to the Fokker-Planck Equation with a Bottomless Action

TL;DR

This paper introduces a novel Langevin equation with a field-dependent kernel for bottomless systems, providing an analytical solution to the associated Fokker-Planck equation and clarifying its relation to the Feynman measure.

Contribution

It presents a new Langevin formulation for bottomless systems and analytically solves the resulting Fokker-Planck equation within stochastic quantization.

Findings

Analytical solution to the Fokker-Planck equation for bottomless systems.

Clarification of the connection to the non-normalizable Feynman measure.

New approach for stochastic quantization of bottomless potentials.

Abstract

A new Langevin equation with a field-dependent kernel is proposed to deal with bottomless systems within the framework of the stochastic quantization of Parisi and Wu. The corresponding Fokker-Planck equation is shown to be a diffusion-type equation and is solved analytically. An interesting connection between the solution with the ordinary Feynman measure, which in this case is not normalizable, is clarified.