Path Integral for Multiplicative Noise: Generalized Fokker-Planck Equation and Entropy Production Rate in Stochastic Processes With Threshold

TL;DR

This work develops a generalized path integral formalism for stochastic systems with arbitrary multiplicative noise by introducing a generalized Itô diffusive process driven by a noise $\eta(t)$ with cumulant generating function $\mathcal{K}_{\eta}$ and a stochastic-calculus parameter $\gamma\in[0,1]$. Using the Parisi–Sourlas supersymmetric approach, it derives a stochastic path integral with a Lagrangian that incorporates $\mathcal K_{\eta}$ and links to Onsager–Machlup and MSRJD functionals in the white-noise limit; the framework yields a forward Fokker–Planck equation FP-GIDP$(\eta,\gamma)$ that reduces to known forms when $\mu$ is proportional to $\sigma$ and when coefficients are homogeneous. The paper then applies the formalism to four thresholded processes—BM, GBM, LF$(\alpha)$, and GLF$(\alpha)$—deriving explicit transition densities $\Psi(x,t)$ and comparing with simulations, and introduces GLF$(\alpha)$ as a new process solvable without Itô’s lemma. Entropy analysis shows that restricted BM and GBM exhibit quasi-steady states with nonzero Shannon entropy production, highlighting non-equilibrium features induced by multiplicative noise and thresholds. The results provide a unifying, noise-structure-agnostic tool for analyzing non-Gaussian stochastic dynamics with thresholds and open pathways for multi-variable extensions and applications across physics, finance, and biology.

Abstract

This paper introduces a comprehensive extension of the path integral formalism to model stochastic processes with arbitrary multiplicative noise. To do so, Itô diffusive process is generalized by incorporating a multiplicative noise term $(η(t))$ that affects the diffusive coefficient in the stochastic differential equation. Then, using the Parisi-Sourlas method, we estimate the transition probability between states of a stochastic variable $(X(t))$ based on the cumulant generating function $(\mathcal{K}_η)$ of the noise. A parameter $γ\in[0,1]$ is introduced to account for the type of stochastic calculation used and its effect on the Jacobian of the path integral formalism. Next, the Feynman-Kac functional is then employed to derive the Fokker-Planck equation for generalized Itô diffusive processes, addressing issues with higher-order derivatives and ensuring compatibility with known functionals such as Onsager-Machlup and Martin-Siggia-Rose-Janssen-De Dominicis in the white noise case. The general solution for the Fokker-Planck equation is provided when the stochastic drift is proportional to the diffusive coefficient and $\mathcal{K}_η$ is scale-invariant. Finally, the Brownian motion ($BM$), the geometric Brownian motion ($GBM$), the Levy $α$-stable flight ($LF(α)$), and the geometric Levy $α$-stable flight ($GLF(α)$) are simulated with thresholds, providing analytical comparisons for the probability density, Shannon entropy, and entropy production rate. It is found that restricted $BM$ and restricted $GBM$ exhibit quasi-steady states since the rate of entropy production never vanishes. It is also worth mentioning that in this work the $GLF(α)$ is defined for the first time in the literature and it is shown that its solution is found without the need for Itô's lemma.

Figures (6)

• Figure 1: Temporal Evolution of the probability density function $\Psi_{BM}(x,t)$ for restricted Brownian Motion with parameters $x_{0}=2$, $t_{0}=0$, $\mu=1\times10^{-1}$, $\sigma=3$, and $x_{V}=1$, using $N_{s}=4\times10^{4}$ trajectories, $N_{t}=5\times10^{3}$ time steps, $N_{b}=2\times10^{2}$ bins, and a final time of $t_{f}=1\times10^{2}$. In all cases, the solid line corresponds to the theoretical fit while the points correspond to the simulated data.
• Figure 2: Temporal Evolution of the probability density function $\Psi_{GBM}(x,t)$ for restricted geometric Brownian Motion with parameters $x_{0}=6\times10^{1}$, $t_{0}=0$, $\mu=2.05\times10^{-1}$, $\sigma=8\times10^{-2}$, and $x_{V}=1$, using $N_{s}=1\times10^{5}$ trajectories, $N_{t}=4\times10^{3}$ time steps, $N_{b}=4\times10^{2}$ bins, and a final time of $t_{f}=4\times10^{1}$. In all cases, the solid line corresponds to the theoretical fit while the points correspond to the simulated data.
• Figure 3: Temporal Evolution of the probability density function $\Psi_{LF}(x,t)$ for restricted Levy $\alpha$-stable flight with parameters $x_{0}=5$, $t_{0}=0$, $\alpha=1.8$, $\beta=0.9$, $\mu=5\times10^{-1}$, $\sigma=4\times10^{-1}$, and $x_{V}=-1$, using $N_{s}=1\times10^{5}$ trajectories, $N_{t}=5\times10^{3}$ time steps, $N_{b}=2\times10^{2}$ bins, and a final time of $t_{f}=5\times10^{1}$. In all cases, the solid line corresponds to the theoretical fit while the points correspond to the simulated data.
• Figure 4: Temporal Evolution of the probability density function $\Psi_{GLF}(x,t)$ for restricted geometric Levy $\alpha$-stable flight with parameters $x_{0}=8\times10^{1}$, $t_{0}=0$, $\alpha=1.9$, $\beta=0.5$, $\mu=1.05\times10^{-1}$, $\sigma=1\times10^{-1}$, and $x_{V}=1$, using $N_{s}=1\times10^{5}$ trajectories, $N_{t}=8\times10^{3}$ time steps, $N_{b}=2\times10^{2}$ bins, and a final time of $t_{f}=4\times10^{1}$. In all cases, the solid line corresponds to the theoretical fit while the points correspond to the simulated data.
• Figure 5: Entropy analysis for restricted Brownian Motion with parameters $x_{0}=2$, $t_{0}=0$, $\mu=1\times10^{-1}$, $\sigma=3$, and $x_{V}=1$, using $N_{s}=4\times10^{4}$ trajectories, $N_{t}=5\times10^{3}$ time steps, $N_{b}=2\times10^{2}$ bins, and a final time of $t_{f}=1\times10^{2}$. (A) Temporal evolution of the Shannon entropy $\mathbb{H}_{1}(\Psi_{BM},t)$. (B) Shannon entropy production rate $\frac{d}{dt}\mathbb{H}_{1}(\Psi_{BM},t)$. In both cases, the red solid line corresponds to the theoretical fits Eq. \ref{['Eq. Entropy Production 17']} and Eq. \ref{['Eq. Entropy Production 19']}, while the green solid line corresponds to the standard BM case (no threshold) and the points correspond to the simulated data.