Erratum to: On the existence of smooth densities for jump processes
Erratum to: On the existence of smooth densities for jump processes
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勘误表：关于跳跃过程的平滑密度的存在性
DOI：
10.1007/s00440-010-0267-x
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
J. Picard
中科院分区：
文献类型：
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作者：
J. Picard
然而，在定理3.1的证明中存在问题。从技术上讲，定理3.1是从定理2.1推导出来的，所以我们必须验证后一个定理的假设是否满足。特别地，我们必须证明条件（2.3）对几乎任何τ∈ A（ρ）k都是成立的。让我们仔细地看一下（3.11）式后面的不等式序列;正如已经说过的，这些不等式应该对几乎任何τ∈ A（ρ）k都成立。第一个不等式是平凡的，因为随机变量R满足R≤ ρ。在第二个不等式中，我们想应用公式3.4。第一个问题是R是随机的，而公式3.4是针对确定性半径而写的。此外，更重要的是，即使R是确定性的，应用式（3.4）也需要τ∈ A（R）k，但这不一定是真的，因为A（R）可以严格包含在A（ρ）中。本文的目的是对定理3.1给出一个正确的说明，并考虑其对以后结果的影响。特别地，推论4.4和4.5仍然有效;推论4.4例如在[2]中使用，因此不被这个修正改变。
However, there is a problem in the proof of Theorem 3.1. Technically, Theorem 3.1 is deduced from Theorem 2.1, so we have to verify that the assumptions of this latter theorem are fulfilled. In particular, we have to prove that condition (2.3) is verified for almost any τ∈ A (ρ) k. Let us look carefully at the sequence of inequalities following (3.11); as it has been said, these inequalities should be valid for almost any τ∈ A (ρ) k. The first inequality is trivial since the random variable R satisfies R≤ ρ. In the second inequality, we want to apply (3.4). A first problem is that R is random whereas (3.4) was written for a deterministic radius. Moreover and more importantly, even if R were deterministic, the application of (3.4) would require τ∈ A (R) k, and this is not necessarily true since A (R) may be strictly included in A (ρ). Thus the proof of the theorem is not correct.Our aim in this note is to give a corrected statement for Theorem 3.1, and to consider the consequences on subsequent results. In particular, Corollaries 4.4 and 4.5 are still valid; Corollary 4.4 is used for instance in [2] which therefore is not changed by this correction.