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 ${\mathcal{L}}_0+r_t^{\epsilon }{\mathcal{L}}_1$ driven by a regenerative random environment. When the integrated activity over a regeneration cycle has a regularly varying tail of index $\beta \in (0,1)$, the cumulative operational clock converges to a $\beta$-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 $\overline{\mathcal{L}}={\mathcal{L}}_0-\kappa (-{\mathcal{L}}_1{)}^{\beta }$. Under uniform ellipticity of ${\mathcal{L}}_1$, the fractional power $(-{\mathcal{L}}_1{)}^{\beta }$ is identified as a $2\beta$-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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