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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.01387 |
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| _version_ | 1866910858956767232 |
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| author | Abril-Bermúdez, F. S. Quimbay, C. J. Trinidad-Segovia, J. E. Sánchez-Granero, M. A |
| author_facet | Abril-Bermúdez, F. S. Quimbay, C. J. Trinidad-Segovia, J. E. Sánchez-Granero, M. A |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01387 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Path Integral for Multiplicative Noise: Generalized Fokker-Planck Equation and Entropy Production Rate in Stochastic Processes With Threshold Abril-Bermúdez, F. S. Quimbay, C. J. Trinidad-Segovia, J. E. Sánchez-Granero, M. A Mathematical Physics 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. |
| title | Path Integral for Multiplicative Noise: Generalized Fokker-Planck Equation and Entropy Production Rate in Stochastic Processes With Threshold |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2410.01387 |