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Let M be a compact C manifold of dimension d. A Coo metric ,, on M gives rise to the notions of a-potential and a-energy, a > 0. These were discussed in [2] and [4] in the context of the asymptotic behaviour of the diffusion on M with generator L A + V, where A is the Laplace operator associated with , and V is a vector field. In this paper we shall continue that discussion, assuming for simplicity V 0 and d > 2. In particular, we shall prove an almost everywhere central limit theorem (a.e. CLT) for the occupation measures of the diffusion, a theorem similar to the one we proved in [5] for liD random variables. The occupation measures assume their values in a nuclear space. We shall exploit "exponential mixing" of the diffusion. As an application of our a.e. CLT, we shall recover the spectrum of the operator A from the development of the a-energy on a typical diffusion path. A classical CLT for the occupation measures can be found in [2]. For background material on a-potentials and a-energy in Ra we refer the reader to [8]. In the case of a compact Riemannian manifold, the a-potential kernels {g, a > 0} were defined in [3] in terms of the fundamental solution of the heat equation. To be precise, if p is the solution of

A simultaneous almost everywhere central limit theorem for diffusions and its application to path energy and eigenvalues of the Laplacian