Stability problems for Cantor stochastic differential equations

We consider driftless stochastic differential equations and the diffusions starting from the positive half line. It is shown that the Feller test for explosions gives a necessary and sufficient condition to hold pathwise uniqueness for diffusion coefficients that are positive and monotonically increasing or decreasing on the positive half line and the value at the origin is zero. Then, stability problems are studied from the aspect of Hölder-continuity and a generalized Nakao–Le Gall condition. Comparing the convergence rate of Hölder-continuous case, the sharpness and stability of the Nakao–Le Gall condition on Cantor stochastic differential equations are confirmed. Furthermore, using the Malliavin calculus, we construct a smooth solution to degenerate second order Fokker–Planck equations under weak conditions on the coefficients.