We continue the work started in Part I of this article, showing how the addition of noise can stabilize an otherwise unstable system. The analysis makes use of nearly optimal Lyapunov functions. In this continuation, we remove the main limiting assum
ption of Part I by an inductive procedure as well as establish a lower bound which shows that our construction is radially sharp. We also prove a version of Peskirs cite{Peskir_07} generalized Tanaka formula adapted to patching together Lyapunov functions. This greatly simplifies the analysis used in previous works.
Motivated by applications to stochastic differential equations, an extension of H{o}rmanders hypoellipticity theorem is proved for second-order degenerate elliptic operators with non-smooth coefficients. The main results are established using point-w
ise Bessel kernel estimates and a weighted Sobolev inequality of Stein and Weiss. Of particular interest is that our results apply to operators with quite general first-order terms.
