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arXiv · 2312.05475

A standard form of master equations for general non-Markovian jump processes: the Laplace-space embedding framework and asymptotic solution

Kiyoshi Kanazawa·Didier Sornette
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Abstract

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.

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BibTeXRIS

Kiyoshi Kanazawa, Didier Sornette. 2023-12-09. A standard form of master equations for general non-Markovian jump processes: the Laplace-space embedding framework and asymptotic solution. https://doi.org/10.1103/physrevresearch.6.023270

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