Diffusion Processes and Diffusion Equations in Random Environment

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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.