Path integrals and stochastic calculus

Abstract

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.

96 pages, 4 figures. New title, expanded introduction and additional references. Version accepted in Advandes in Physics

References in corpus (10)

• Motility-Induced Phase Separation
• State-dependent diffusion: thermodynamic consistency and its path integral formulation
• Mapping out of equilibrium into equilibrium in one-dimensional transport models
• Symmetries of generating functionals of Langevin processes with colored multiplicative noise
• Field theories and exact stochastic equations for interacting particle systems
• Breakdown of the coherent state path integral: two simple examples
• Hidden symmetries and equilibrium properties of multiplicative white-noise stochastic processes
• Universal Form of Stochastic Evolution for Slow Variables in Equilibrium Systems
• Dynamical mean-field theory: from ecosystems to reaction networks
• Stochastic action for tubes: Connecting path probabilities to measurement

Cited by in corpus (8)

• Quantum Effects on the Synchronization Dynamics of the Kuramoto Model
• Evaluation of transition rates from nonequilibrium instantons
• Fluctuation Dissipation Relations for Active Field Theories
• Large deviations at level 2.5 and for trajectories observables of diffusion processes : the missing parts with respect to their random-walks counterparts
• On the integration of Ito equations with a random or a W-symmetry
• Link between Continuous and Discrete Descriptions of Noise in Nonlinear Resistive Electrical Components
• Characteristic oscillations in frequency-resolved heat dissipation of linear time-delayed Langevin systems: Approach from the violation of the fluctuation-response relation
• Consistent expansion of the Langevin propagator with application to entropy production