ABSTRACT

In this article, we are interested in studying the transition density function of the couple given by the first hitting time of a fixed threshold by a one-dimensional uniformly elliptic diffusion process and the associated stopped process. Our strategy relies on the parametrix technique applied to the related semigroup. It notably allows us to extend standard PDEs results on Green and Poisson functions, see e.g. Garroni and Menaldi [Green functions for second order parabolic integro-differential problems, volume 275 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992], by establishing the infinite expansion of the corresponding transition density for irregular coefficients, here bounded measurable drift coefficient and bounded η-Hölder-continuous diffusion coefficient. As a by-product, we establish some Gaussian upper estimates.

抽象的

在本文中，我们有兴趣研究由一维均匀椭圆扩散过程和相关的停止过程产生的，由固定阈值的首次命中时间给出的偶对的跃迁密度函数。我们的策略依赖于应用于相关半群的parametrix技术。值得注意的是，它可以使我们扩展有关Green和Poisson函数的标准PDE结果，请参见例如Garroni和Menaldi [二阶抛物线积分微分问题的Green函数，Pitman研究笔记在数学系列中的第275卷，朗文科学技术公司，哈洛；[通过与John Wiley＆Sons，Inc.在纽约共同出版，1992年]，通过建立不规则系数的相应跃迁密度的无限扩展，这里有界可测漂移系数和有界η -Hölder连续扩散系数。作为副产品，我们建立了一些高斯上限估计。