Research on properties of Markov processes governed by the pseudo-differential operators with variable orders and application of the m to nonlinear analysis

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As is well known, under suitable conditions, it has been shown that there exist pure jump type Markov processes governed by Levy. generating operators with degenerate Levy mesures. So we would like to know what conditions these Markov processes have their transition densities under. Recently, by using MALLIAVIN calculus, Kunita has constructed transition densities of these Markov proceses in some class. So, we tried to adapt the pseudo-differential operators theory for this problem and restricted our study to the case that the supports of Levy measures degenerated into mutualy independent d lines for each x in RィイD1dィエD1. Cosequently, we have got that Markov processes governed the following generators, L have transition densities. The L is<<numerical formula>>where θィイD2jィエD2(x) (j = 1, 2,…, d) are smooth RィイD1dィエD1-valued functions with bounded derivatives on RィイD1dィエD1 and satisfy |θィイD2jィエD2(x)|=1(j = 1, 2,…, d). Putting Θ(x)=(θィイD21ィエD2(x), θィイD22ィエD2(x), …, θィイD2dィエD2(x)), we assume that the eigenvalues of Θ(x)*Θ(x) are unifomly bounded to the below. And also, α is a constant satisfying 1 < α < 2 and nィイD2jィエD2(x,y) (j = 1,…, d) are smooth funcutions with bounded derivatives satisfying usual coditions. Now, we are rounding off the above work. We regret to say that we were able to have no result about the relation between nolinear differential operators and stochastic processes. But while we were studing this problem, we had the following results.(1) A one dimensional hyperbolic equation uィイD2ttィエD2 - uィイD2xxィエD2 = 0 is treated under a free boundary condition uィイD32(/)XィエD3-uィイD32(/)tィエD3=QィイD12ィエD1. The existance and the uniqueness of a classical solution is established loccaly.(2) A weak solution to some forth order nonlinear parabolic equation is constructed by the method of time semidisceretization. A technique of geometoric measure theory is employed in order to obtain to obtain the convergence of the nonlinear terms.