Small Ball Probabilities And A Support Theorem For The Stochastic Heat Equation

We consider the following stochastic partial differential equation on t >= 0, x is an element of [0, J], J >= 1, where we consider [0, J] to be the circle with end points identified,partial derivative(t)u(t, x) = 1/2 partial derivative(2)(x) u(t, x) + g(t, x, u) + sigma(t, x, u) (W) over dot (t, x),(W) over dot (t, x) is 2-parameter d-dimensional vector valued white noise and sigma is function from R+ x R x R-d to space of symmetric d x d matrices which is Lipschitz in u. We assume that sigma is uniformly elliptic and that g is uniformly bounded. Assuming that u(0, x) = 0, we prove small ball probabilities for the solution u. We also prove a support theorem for solutions, when u(0, x) is not necessarily zero.