We show convergence of a finite difference scheme to solve the abstract Cauchy problem on a scale of Banach spaces which includes that for Kowalevskaya’s system. We show convergence of consistent difference schemes even for unstable cases.

We classify the symmetric solutions for the two dimensional degenerate Garnier system G(5) which is a generalization of the second Painlevé equation. We calculate the linear monodromy of the symmetric solutions explicitly.

We give recurrence/transience criteria for skew products of one dimensional diffusion process and the spherical Brownian motion with respect to a positive continuous additive functional of the former one dimensional diffusion process. Further we give recurrence/transience criteria for their time changed processes.