The heat equation with Lévy noise1

We prove short-time existence for parabolic equations with Lévy noise of the form ut=Δαu+uγL̇(t, x), t>0, x∈D⊂Rdu(t, x)=0 for x∈Dc,u(0, x)=u0(x),where L̇ is nonnegative Lévy noise of index p∈(0, 1),Δα is the α/2 power of the Laplacian, α∈(0,2],γ>0, and u0(x) is a continuous nonnegative function. D is a bounded open domain in Rd. A sufficient condition for short time existence is d<(1−p)αγp−(1−p).While we cannot prove uniqueness, we show that the solution we construct is minimal among all solutions.