Strong invariance principle for singular diffusions

We study a singular diffusion on Euclidean space which is characterized by the solution of a classical Itô stochastic differential equation in which the diffusion coefficient is not necessarily of full rank. Our motivation is in earlier results of Basak (J. Multivariate Anal. 39 (1991) 44) and Basak and Bhattacharya (Ann. Probab. 20 (1992) 312), who establish sufficient conditions for singular diffusions to have a unique invariant probability and obtain a functional central limit theorem and functional law of the iterated logarithm for a large class of real-valued functions of the diffusion. Under similar conditions we establish a strong invariance principle for vector-valued functions of the diffusion, and use this to derive several asymptotic properties of the singular diffusion, including upper/lower-function estimates and a vector form of the functional law of the iterated logarithm.