# Making Birth-Death Processes from Backward Fokker-Planck Equations for Computing Expectations in Langevin Systems

## TL;DR

The paper introduces a novel method using birth-death processes derived from backward Fokker-Planck equations to efficiently compute expectations in Langevin systems, reducing computational effort for multiple initial conditions.

## Contribution

It presents a new approach combining dummy variables and Itô calculus to derive birth-death processes for expectation evaluation in Langevin systems, enabling single integration for multiple scenarios.

## Key findings

- Method reduces computational cost by enabling one-time integration.
- Applicable to various initial conditions and function centers.
- Demonstrated on a double-well system with sigmoid functions.

## Abstract

A method to direct evaluation of expectations for Langevin systems (stochastic differential equations) is proposed. The method is based on a birth-death process which is derived using combinations of dummy variables and It{\^o} formula. As a pedagogical example, a double-well system and expectations for sigmoid-type functions are used. It is shown that the proposed method has some merits from computational point of view; only one time-integration for the birth-death process gives expectations for various initial conditions in the original Langevin systems. Furthermore, the same time-integration result is available for computing various center positions of the sigmoid-type functions.