Jump limits of continuous multiscale diffusions: Homogenization under heavy-tailed activity bursts
Abstract
We show that purely continuous microscopic diffusions can homogenize to a Markov process with genuine jumps. Consider a multiscale diffusion with time-dependent generator ℒ0+rtεℒ1 driven by a regenerative random environment. When the integrated activity over a regeneration cycle has a regularly varying tail of index β∈(0,1), the cumulative operational clock converges to a β-stable subordinator, compressing finite activity-driven diffusion into vanishingly short physical time intervals. To overcome the noncommutativity of ℒ0 and ℒ1, we develop an extended-time parametrization paired with Khasminskii-type time discretization, establishing finite-dimensional convergence and strong semigroup convergence on C0(ℝd) toward a jump process with nonlocal generator ℒ̅=ℒ0−κ(−ℒ1)β. Under uniform ellipticity of ℒ1, the fractional power (−ℒ1)β is identified as a 2β-stable-like nonlocal operator. We also clarify physical-time path-topology obstructions and apply the framework to Poisson resetting pulses, Ornstein--Uhlenbeck diffusions, and multiscale stochastic volatility.
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BibTeX
@misc{zhang2026jump,
title = {{Jump limits of continuous multiscale diffusions: Homogenization under heavy-tailed activity bursts}},
author = {Qi Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20302},
eprint = {2610.20302},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20302},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Jump limits of continuous multiscale diffusions: Homogenization under heavy-tailed activity bursts
AU - Qi Zhang
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20302
UR - https://arxiv.org/abs/2610.20302
AB - We show that purely continuous microscopic diffusions can homogenize to a Markov process with genuine jumps. Consider a multiscale diffusion with time-dependent generator ℒ_0+r_t^εℒ_1 driven by a regenerative random environment. When the integrated activity over a regeneration cycle has a regularly varying tail of index β∈(0,1), the cumulative operational clock converges to a β-stable subordinator, compressing finite activity-driven diffusion into vanishingly short physical time intervals. To overcome the noncommutativity of ℒ_0 and ℒ_1, we develop an extended-time parametrization paired with Khasminskii-type time discretization, establishing finite-dimensional convergence and strong semigroup convergence on C_0(ℝ^d) toward a jump process with nonlocal generator ℒ̅=ℒ_0−κ(−ℒ_1)^β. Under uniform ellipticity of ℒ_1, the fractional power (−ℒ_1)^β is identified as a 2β-stable-like nonlocal operator. We also clarify physical-time path-topology obstructions and apply the framework to Poisson resetting pulses, Ornstein--Uhlenbeck diffusions, and multiscale stochastic volatility.
ER -