Sharp estimation of the almost-sure Lyapunov exponent for the Anderson model in continuous space

In this article we study the exponential behavior of the continuous stochastic Anderson model, i.e. the solution of the stochastic partial differential equation u(t, x) = 1 + f(0)(t) kappa Delta(x)u(s,x) ds+ f(0)(t) W(ds,x) u (s,x), when the spatial parameter x is continuous, specifically x is an element of R, and W is a Gaussian field on R+ x R that is Brownian in time, but whose spatial distribution is widely unrestricted. We give a partial existence result of the Lyapunov exponent defined as lim(t ->infinity) t(-1) log u( t, x). Furthermore, we find upper and lower bounds for lim sup(t ->infinity) t(-1) log u(t, x) and lim inf(t ->infinity) t(-1) log u(t, x) respectively, as functions of the diffusion constant. which depend on the regularity of W in x. Our bounds are sharper, work for a wider range of regularity scales, and are significantly easier to prove than all previously known results. When the uniform modulus of continuity of the process W is in the logarithmic scale, our bounds are optimal.