Exact Solvability of Some Spdes

These lecture notes are based on Ivan Corwin’s summer 2014 MSRI summer school on SPDEs, as well as Jeffrey Kuan’s TA sessions accompanying these lectures. Please email [email protected] if you have questions or find mistakes. 1. Mild solution to the stochastic heat equation The stochastic heat equation (SHE) with multiplicative noise looks (in differential form) like { ∂tz = 1 2∂xxz + zξ z(0, x) = z0(x) where z : R+×R → R and z0 is (possibly random) initial data which is independent of the white noise ξ. Recall that formally ξ has covariance E [ ξ(t, x)ξ(s, y) ] “ = ”δt=sδx=y, though this is only true in a weak, or integrated sense. See Section 10 for background on ξ. The noise ξ is constructed on a probability space L2(Ω,F ,P). We would like to ultimately consider z0(x) = δx=0 initial data. We will start, however, with L2(Ω,F ,P) bounded initial data and prove uniqueness and existence (and state regularity / positivity results without proofs). We will also state (without proof) a more general class of solutions considered by Bertini-Cancrini [5]. In the next section we will explain a different way to construct a solution for δx=0 initial data via chaos series. Definition 1.1. A mild solution to the SHE satisfies for all t > 0, x ∈ R the Duhamel form equation z(t, x) = ∫ R p(t, x− y)z0(y)dy + ∫ t