Large-time and small-ball asymptotics for quadratic functionals of Gaussian diffusions

Using asymptotic analysis of the Laplace transform, we establish almost sure divergence of certain integrals and derive logarithmic asymptotic of small ball probabilities for quadratic forms of Gaussian diffusion processes. The large time behavior of the quadratic forms exhibits little dependence on the drift and diffusion matrices or the initial conditions, and, if the noise driving the equation is not degenerate, then similar universality also holds for small ball probabilities. On the other hand, degenerate noise leads to a variety of different asymptotics of small ball probabilities, including unexpected influence of the initial conditions. AMS Subject Classification 60G15, 60J60.