  Given a symmetric diffusion process and a jump process on the same underlying
space, is there a subordinator such that the jump process and the subordinated
diffusion processes are comparable? We address this question when the diffusion
satisfies a sub-Gaussian heat kernel estimate and the jump process satisfies a
polynomial-type jump kernel bounds. Under these assumptions, we obtain
necessary and sufficient conditions on the jump kernel estimate for such a
subordinator to exist. As an application of our results and the recent
stability results of Chen, Kumagai, and Wang, we obtain parabolic Harnack
inequality for a large family of jump processes. In particular, we show that
any jump process with polynomial-type jump kernel bounds on such a space
satisfies the parabolic Harnack inequality.
