For multidimensional Lévy processes with finite jump measure outside the Gaussian range, Hunt's hypothesis is equivalent to the same hypothesis holding for every one-dimensional projection, and new energy decay criteria are proved for products of independent processes.
Two Theorems on Hunt's Hypothesis (H) for Markov Processes
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abstract
Hunt's hypothesis (H) and the related Getoor's conjecture is one of the most important problems in the basic theory of Markov processes. In this paper, we investigate the invariance of Hunt's hypothesis (H) for Markov processes under two classes of transformations, which are change of measure and subordination. Our first theorem shows that for two standard processes $(X_t)$ and $(Y_t)$, if $(X_t)$ satisfies (H) and $(Y_t)$ is locally absolutely continuous with respect to $(X_t)$, then $(Y_t)$ satisfies (H). Our second theorem shows that a standard process $(X_t)$ satisfies (H) if and only if $(X_{\tau_t})$ satisfies (H) for some (and hence any) subordinator $(\tau_t)$ which is independent of $(X_t)$ and has a positive drift coefficient. Applications of the two theorems are given.
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2019 1verdicts
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Hunt's Hypothesis (H) for Markov Processes: Survey and Beyond
For multidimensional Lévy processes with finite jump measure outside the Gaussian range, Hunt's hypothesis is equivalent to the same hypothesis holding for every one-dimensional projection, and new energy decay criteria are proved for products of independent processes.