Jump limits of continuous multiscale diffusions: Homogenization under heavy-tailed activity bursts

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 regener