Asymptotic properties of asymptotically homogeneous diffusion processes on a compact manifold (Q1070660)

Let \(\{\xi(t),P_{s,x}\}\) be a time inhomogeneous diffusion with generator \(L_ t=2^{-1}a^{ij}(t,x)\partial^ 2/\partial x^ i\partial j^ j+b^ i(t,x)\partial/\partial x^ i\) and \(\{\lambda(t),P_ x\}\) be a homogeneous diffusion with generator \(L=2^{-1}a^{ij}(x)\partial^ 2/\partial x^ i\partial x^ j+b^ i(x)\partial /\partial x^ i\) such that for every smooth \(\phi\), \(L_ t\phi \to_{t\to \infty}L\phi\) uniformly on compact sets - this is called asymptotic homogeneity. Results: With \(P_{s,x}\) probability one, the weak* limit points for \(t\to \infty\) of the ''occupation distributions'' \((t-s)^{- 1}\int^{t}_{s}\delta_{\xi (\tau)}d\tau\) are invariant probabilities of \(\lambda(t)\) (strong law of large numbers), furthermore a central limit theorem and theorems on large deviations are proved.