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

Kiyoshi Kanazawa and 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.

Date: 2023-12
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Published in Phys. Rev. Res. 6, 023270 (2024)

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