Path integrals and stochastic calculus

doi:10.1080/00018732.2023.2199229

Path integrals and stochastic calculus

2023-04-19

13

OA

AI

T

Thibaut Arnoulx de Pirey *

L

Leticia F. Cugliandolo

DOI：10.1080/00018732.2023.2199229

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Path integrals are a ubiquitous tool in theoretical physics. However, their use is sometimes hindered by the lack of control on various manipulations-such as performing a change of the integration path-one would like to carry out in the light-hearted fashion that physicists enjoy. Similar issues arise in the field of stochastic calculus, which we review to prepare the ground for a proper construction of path integrals. At the level of path integration, and in arbitrary space dimension, we not only report on existing Riemannian geometry-based approaches that render path integrals amenable to the standard rules of calculus, but also bring forth new routes, based on a fully time-discretized approach, that achieve the same goal. We illustrate these various definitions of path integration on simple examples such as the diffusion of a particle on a sphere.

Keywords:

theoretical physics

path integrals

stochastic calculus

Path integrals and stochastic calculus

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