Rules of calculus in the path integral representation of white noise Langevin equations: the Onsager–Machlup approach

Abstract

The definition and manipulation of Langevin equations with multiplicative white noise require special care (one has to specify the time discretisation and a stochastic chain rule has to be used to perform changes of variables). While discretisation-scheme transformations and non-linear changes of variable can be safely performed on the Langevin equation, these same transformations lead to inconsistencies in its path-integral representation. We identify their origin and we show how to extend the well-known Itō prescription (dB 2 = dt) in a way that defines a modified stochastic calculus to be used inside the path-integral representation of the process, in its Onsager-Machlup form.

Origins and diagnostics of the nonequilibrium character of active systems

“…The first step in calculating the entropy production rate and all the subsequent analysis consists in writing the path probability of the particle. Since the noises in both (8) and (9) are Gaussian, the path probability weight [65] takes the form P = e −A , where, for the inertia-less limit of the equations of motion (8) and 9, the stochastic action in Onsager-Machlup [65,66] form is…”

Markov blankets, information geometry and stochastic thermodynamics

“…The key to this is to move from thinking about a density over a particle's current location to quantifying the probability of it having followed a given path. This is given by the following (Stratonovich 3 ) path integral [26]: normalℑfalse(xfalse[τfalse]false)=14italicΓ∫ot(x˙⋅x˙−2x˙⋅f+f⋅f+2italicΓ∇⋅f) dτ.…”

A Framework of Nonequilibrium Statistical Mechanics. I. Role and Types of Fluctuations

“…(9) should put initial and final times on an equal footing. Interpretation rules for functional integrals have been developed in [51,52]. The need to introduce interpretation rules for functional integrals should not keep us from taking advantage of this powerful tool, as is well known from a similar situation in the theory of stochastic differential equations.…”

“…The integral in the exponent of eq. (9) encodes the sequential transition probabilities of a Markov process (see [51,52] or Section 11.1 of [58]). Therefore, the infinitesimal generator Q of this Markov process, which is defined as the operator…”

“…(20) corresponds to a forward Euler, or nonanticipating, definition of the functional integral in eq. (9) (see [51,52] or Section 11.1 of [58]). The simplest way to achieve a symmetric treatment of initial and final times in a time step τ is by using the following replacement in the numerator and denominator of eq.…”