Let $p_t(x)$, $f_t(x)$ and $q_t^*(x)$ be the densities at time $t$ of a real Levy process, its running supremum and the entrance law of the reflected excursions at the infimum. We provide relationships between the asymptotic behaviour of $p_t(x)$, $f_t(x)$ and $q_t^*(x)$, when $t$ is small and $x$ is large. Then for large $x$, these asymptotic behaviours are compared to this of the density of the Levy measure. We show in particular that, under mild conditions, if $p_t(x)$ is comparable to $t\nu(x)$, as $t\rightarrow0$ and $x\rightarrow\infty$, then so is $f_t(x)$.

中文翻译：

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与 Lévy 过程相关的密度行为

令 $p_t(x)$、$f_t(x)$ 和 $q_t^*(x)$ 是一个实际 Levy 过程在时间 $t$ 的密度、它的运行上限值和反射偏移的入口定律下限。我们提供了 $p_t(x)$、$f_t(x)$ 和 $q_t^*(x)$ 的渐近行为之间的关系，当 $t$ 小而 $x$ 大时。然后对于较大的 $x$，将这些渐近行为与 Levy 测度的密度进行比较。我们特别表明，在温和条件下，如果 $p_t(x)$ 与 $t\nu(x)$ 相当，如 $t\rightarrow0$ 和 $x\rightarrow\infty$，那么 $f_t( x)$。