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 $\mathcal{L}_{0}+r_{t}^ε\mathcal{L}_{1}$ driven by a regenerative random environment. When the integrated activity over a regeneration cycle has a regularly varying tail of index $β\in(0,1)$, 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 $\mathcal{L}_0$ and $\mathcal{L}_1$, we develop an extended-time parametrization paired with Khasminskii-type time discretization, establishing finite-dimensional convergence and strong semigroup convergence on $C_{0}(\mathbb{R}^{d})$ toward a jump process with nonlocal generator $\overline{\mathcal{L}}=\mathcal{L}_{0}-κ(-\mathcal{L}_{1})^β$. Under uniform ellipticity of $\mathcal{L}_{1}$, the fractional power $(-\mathcal{L}_{1})^β$ 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.