Complex paths for real stochastic processes

TL;DR

This paper revisits the calculation of decay rates in metastable states within stochastic processes, resolving previous mathematical difficulties by using extremal solutions in the Ito path integral formulation.

Contribution

It introduces a method using extremal solutions in the Ito formulation to accurately compute decay rates, improving mathematical rigor over previous approaches.

Findings

Resolved mathematical difficulties in decay rate calculations

Demonstrated the method with a simple potential example

The mechanism applies broadly beyond the example used

Abstract

The calculation of the decay rate of a metastable state in the path-integral formulation of stochastic processes is revisited. Previous derivations of this rate were achieved at the cost of a step that is difficult to justify mathematically. We show that this difficulty can be resolved by working with an extremal solution that arises naturally in the Ito formulation of the path integral. To make the analysis as transparent as possible, we choose a simple potential for which the extremal solution can be written in terms of elementary functions. The mechanism identified here, however, is not restricted to this example and holds more generally.