A standard form of master equations for general non-Markovian jump processes: the Laplace-space embedding framework and asymptotic solution
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| Format: | Preprint |
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2023
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| _version_ | 1866917691958231040 |
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| author | Kanazawa, Kiyoshi Sornette, Didier |
| author_facet | Kanazawa, Kiyoshi Sornette, Didier |
| contents | We present a standard form of master equations (ME) for general one-dimensional non-Markovian (history-dependent) jump processes, complemented by an asymptotic solution derived from an expanded system-size approach. The ME is obtained by developing a general Markovian embedding using a suitable set of auxiliary field variables. This Markovian embedding uses a Laplace-convolution operation applied to the velocity trajectory. We introduce an asymptotic method tailored for this ME standard, generalising the system-size expansion for these jump processes. Under specific stability conditions tied to a single noise source, upon coarse-graining, the Generalized Langevin Equation (GLE) emerges as a universal approximate model for point processes in the weak-coupling limit. This methodology offers a unified analytical toolset for general non-Markovian processes, reinforcing the universal applicability of the GLE founded in microdynamics and the principles of statistical physics. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_05475 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A standard form of master equations for general non-Markovian jump processes: the Laplace-space embedding framework and asymptotic solution Kanazawa, Kiyoshi Sornette, Didier Statistical Mechanics Data Analysis, Statistics and Probability Trading and Market Microstructure We present a standard form of master equations (ME) for general one-dimensional non-Markovian (history-dependent) jump processes, complemented by an asymptotic solution derived from an expanded system-size approach. The ME is obtained by developing a general Markovian embedding using a suitable set of auxiliary field variables. This Markovian embedding uses a Laplace-convolution operation applied to the velocity trajectory. We introduce an asymptotic method tailored for this ME standard, generalising the system-size expansion for these jump processes. Under specific stability conditions tied to a single noise source, upon coarse-graining, the Generalized Langevin Equation (GLE) emerges as a universal approximate model for point processes in the weak-coupling limit. This methodology offers a unified analytical toolset for general non-Markovian processes, reinforcing the universal applicability of the GLE founded in microdynamics and the principles of statistical physics. |
| title | A standard form of master equations for general non-Markovian jump processes: the Laplace-space embedding framework and asymptotic solution |
| topic | Statistical Mechanics Data Analysis, Statistics and Probability Trading and Market Microstructure |
| url | https://arxiv.org/abs/2312.05475 |