{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/path-integral-for-multiplicative-noise","title":"Path Integral for Multiplicative Noise: Generalized Fokker-Planck Equation and Entropy Production Rate in Stochastic Processes With Threshold","arxiv_id":"2410.01387","date":"2024-10-02","proceeding":null,"authors":["F. S. Abril-Bermúdez","C. J. Quimbay","J. E. Trinidad-Segovia","M. A Sánchez-Granero"],"abstract":"This paper introduces a comprehensive extension of the path integral formalism to model stochastic processes with arbitrary multiplicative noise. To do so, It\\^o diffusive process is generalized by incorporating a multiplicative noise term $(\\eta(t))$ 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 $(\\mathcal{K}_{\\eta})$ of the noise. A parameter $\\gamma\\in[0,1]$ 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 It\\^o 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 $\\mathcal{K}_{\\eta}$ is scale-invariant. Finally, the Brownian motion ($BM$), the geometric Brownian motion ($GBM$), the Levy $\\alpha$-stable flight ($LF(\\alpha)$), and the geometric Levy $\\alpha$-stable flight ($GLF(\\alpha)$) 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 $GLF(\\alpha)$ is defined for the first time in the literature and it is shown that its solution is found without the need for It\\^o's lemma.","url_abs":"https://arxiv.org/abs/2410.01387v2","url_pdf":"https://arxiv.org/pdf/2410.01387v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"path-integral-for-multiplicative-noise","repo_url":"https://github.com/fsabrilb/fokker_planck_distribution","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}