Lecture notes for Introduction to SPDE , Spring 2016

with a highly irregular function F (t, x), as well as nonlinear versions of these equations. However, if F is ”very rough” then, presumably, the solution u(t, x) will not be very smooth either, hence one would not be able to differentiate it in time or space, and the sense in which u(t, x) solves the corresponding equation is not quite clear. It is natural to think of u(t, x) as a weak solution of the PDE as that would not require differentiation – but if the PDE (unlike the examples above) is nonlinear we would still need to know that u(t, x) is a function, which is not a priori obvious if the force F (t, x) is too irregular. As we will see, it is often the case that u(t, x) is actually not a function. Another obvious issue is to understand what we would mean by a ”rough force”. If, say, (1.1) were posed on the lattice Z, and ∆ were a discrete Laplacian, then (1.1) would be an infinite system of SDE’s at each lattice site, with F (t, x) independent for each site x ∈ Z. This makes clear sense, as long as x is discrete. In order to define this in R, we need to develop some basics. A natural way to generalize independence at each site is to require that F (t, x) is a stationary in time and space mean-zero process such that the two-point correlation function is