Diffusion Processes and Diffusion Equations in Random Environment

基于布朗运动的随机微分方程有大量的研究工作。利用马利文演算阐明了解律存在光滑密度的条件。在本研究中，我们将注意力限制在基于Levy过程的具有跳跃的SDE，并研究了解的定律具有（光滑）密度的条件。对于方程，我们研究了由有限个向量场和同维Levy过程产生的正则SDE。首先，我们证明了如果向量场和Levy过程都是非简并的，该定律具有光滑密度。然后证明了在向量场可能退化但满足霍曼德条件的情况下，该定律具有密度函数。这些结果是Malliavin和Kusuoka-Stroock的工作的延伸，他们研究了布朗运动驱动的SDE的光滑密度的存在。为了证明，我们需要在维纳空间和泊松空间的乘积上使用马利亚文微积分。统一了泊松空间上的Picard方法和维纳空间上的Malliavin方法，进一步得到了积空间上随机变量律具有光滑密度的判据。该判据包括维纳空间上的Malliavin判据和泊松空间上的Picard判据作为特例。我们将该判据应用于具有跳跃的SDE的解，并证明了解的密度律的存在性。

There are extensive works on SDE (stochastic differential equation) based on Brownian motions. Conditions for the existence of the smooth density for the law of the solution have been clearified by using the Malliavin calculus. In this research, we restricted our attention to SDE with jumps based on Levy process and investigated the condition such that the law of the solution has a(smooth) density. As to the equation, we studied the canonical SDE generated by a finite number of vector fields and the same dimensional Levy processes. First, we showed that the law has a smooth density if both the vector fields and Levy processes are nondegenerate. Then we proved that the law has a density function in the case where the vector fields may be degenearate but satisfy Hormanders condition. These results are extensions of the works by Malliavin and Kusuoka-Stroock, who studied the existence of the smooth density in the case of a SDE driven by a Brownian motion.For the proof, we need the Malliavin calculus on the product of the Wiener space and the Poisson space. We unified the Picard's approach on the Poisson space and the Malliavin's approach on the Wiener space and further we obtained a criterion that the law of the random variable on the product space has a smooth density. The criterion includes Malliavin's on the Wiener space and Picard's on the Poisson space as special cases. We applied the criterion to the solution of SDE with jumps and proved the existence of the density for the law of the solution.