论文标题
在将概率符号扩展到其最一般类别的奇异函数的ro。
On the rôle of singular functions in extending the probabilistic symbol to its most general class
论文作者
论文摘要
概率符号是与随机过程的一维边缘相对应的特征函数的右侧衍生物。只要存在衍生物，该对象就会提供有关随机过程的关键信息。对于Lévy过程，一个人获得了特征指数，而（丰富的）过程的符号与经典符号相吻合，这是从伪差算子的理论中众所周知的。将这些课程留在后面，符号仍然存在的最通用的过程是莱维型过程。在马尔可夫流程的框架内，是否可以进一步概括，这是一个悬而未决的问题。我们在本文中回答了这个问题：在狩猎半明天的类别中，莱维型过程正是概率符号存在的过程。留下准连续性，可以构建符号的过程。但是，我们表明，这些过程可能会丢失符号的适用性。令人惊讶的是，在我们的证明中，与某些单数功能相对应的上和下二衍生物起着重要的作用。
The probabilistic symbol is the right-hand side derivative of the characteristic functions corresponding to the one-dimensional marginals of a stochastic process. This object, as long as the derivative exists, provides crucial information concerning the stochastic process. For a Lévy process, one obtains the characteristic exponent while the symbol of a (rich) Feller process coincides with the classical symbol which is well known from the theory of pseudodifferential operators. Leaving these classes behind, the most general class of processes for which the symbol still exists are Lévy-type processes. It has been an open question, whether further generalizations are possible within the framework of Markov processes. We answer this question in the present article: within the class of Hunt semimartingales, Lévy-type processes are exactly those for which the probabilistic symbol exists. Leaving quasi-continuity behind, one can construct processes admitting a symbol. However, we show, that the applicability of the symbol might be lost for these processes. Surprisingly, in our proofs the upper and lower Dini derivatives corresponding to certain singular functions play an important rôle.
