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

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  • Kiyoshi Kanazawa
  • Didier Sornette

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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  • Kiyoshi Kanazawa & Didier Sornette, 2023. "A standard form of master equations for general non-Markovian jump processes: the Laplace-space embedding framework and asymptotic solution," Papers 2312.05475, arXiv.org.
    • Handle: RePEc:arx:papers:2312.05475
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    File URL: http://arxiv.org/pdf/2312.05475
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