Brownian-Time Processes: The PDE Connection and the Half-Derivative Generator
Brownian-Time Processes: The PDE Connection and the Half-Derivative Generator
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DOI：
10.1214/aop/1015345772
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发表时间：
2001-10
期刊：
影响因子：
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通讯作者：
Hassan Allouba;Weian Zheng
中科院分区：
文献类型：
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作者：
Hassan Allouba;Weian Zheng
我们引入了一类有趣的随机过程的基础上布朗时间过程。这些都是通过采取马尔可夫过程，并取代布朗运动的模的时间参数。他们推广的迭代布朗运动（IBM）的Burdzy和马尔可夫蛇的乐加尔，他们介绍了新的有趣的例子。在定义布朗时间过程后，我们将其与四阶抛物偏微分方程联系起来。然后，我们研究他们的退出问题，因为他们退出好域在$\Rd$，并将其连接到椭圆偏微分方程。我们表明，这些过程具有独特的属性，他们解决四阶抛物偏微分方程，但他们的出口分布-至少在标准的布朗时间过程的情况下-解决了通常的二阶狄利克雷问题。我们恢复四阶偏微分方程在椭圆设置编码的布朗时间过程的迭代性质，通过其退出时间，在一个标准的布朗运动。我们还表明，它是可能的，以分配一个正式的发电机，这些非马尔可夫过程，通过给这样一个发电机在半导数的意义。
We introduce a class of interesting stochastic processes based on Brownian-time processes. These are obtained by taking Markov processes and replacing the time parameter with the modulus of Brownian motion. They generalize the iterated Brownian motion (IBM) of Burdzy and the Markov snake of Le Gall, and they introduce new interesting examples. After defining Brownian-time processes, we relate them to fourth order parabolic PDEs. We then study their exit problem as they exit nice domains in $\Rd$, and connect it to elliptic PDEs. We show that these processes have the peculiar property that they solve fourth order parabolic PDEs, but their exit distribution - at least in the standard Brownian-time process case - solves the usual second order Dirichlet problem. We recover fourth order PDEs in the elliptic setting by encoding the iterative nature of the Brownian-time process, through its exit time, in a standard Brownian motion. We also show that it is possible to assign a formal generator to these non-Markovian processes by giving such a generator in the half-derivative sense.