Suppose that Y_t is a d-dimensional symmetric Levy process such that its Levy measure differs from the Levy measure of the isotropic α-stable process (0 < α < 2) by a finite signed measure. For a bounded Lipschitz open set D we compare the Green functions of the process Y with those of its stable counterpart, and we prove several comparability results, both one-sided and two-sided. In particular, assuming an additional condition about the difference between the densities of the Levy measures, namely that it is of the order of |x|~(-d+g) as |x| → 0, where g > 0, we prove that the Green functions are comparable, provided D is connected.rnThese results apply, for example, to the relativistic a-stable process. The bounds for its Green functions were previously known for d > α and smooth sets. Here we consider also the one-dimensional case for α ≥ 1, and we prove that the Green functions for a bounded open interval are comparable, a case that, to the best of our knowledge, had not been treated in the literature.
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机译：假设Y_t是d维对称Levy过程，以致其Levy量度与各向同性α稳定过程的Levy量度（0 <α<2）相差一个有限符号。对于有界的Lipschitz开放集D，我们将过程Y的Green函数与其稳定的对等函数的Green函数进行了比较，并且我们证明了一些可比较的结果，既有单方面的又有双面的。特别地，假设关于征征度量的密度之差的附加条件，即，作为| x |约为| x |〜（-d + g）。 →0，其中g> 0，我们证明，只要D被连接，格林函数是可比的。这些结果例如适用于相对论的a稳定过程。以前，已知d>α和光滑集的格林函数范围。在这里，我们还考虑了α≥1的一维情况，并且证明了有界开放区间的格林函数是可比较的，就我们所知，这种情况在文献中尚未得到处理。
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