Pathwise uniqueness for an SPDE with Holder continuous coefficient driven by α-stable noise

In this paper we study the pathwise uniqueness of nonnegative solution to the following stochastic partial differential equation with Holder continuous noise coefficient: partial derivative X-t(x)/partial derivative t = 1/2 Delta X-t(x) + G(X-t(x)) + H(X-t-(x)) L-t(x), t > 0, x is an element of R, where for i < alpha < 2 and 0 < beta < 1, L denotes an alpha-stable white noise on R+ x R without negative jumps, G satisfies a condition weaker than Lipschitz and H is nondecreasing and beta-Holder continuous. For G equivalent to 0 and H(x) = x(beta), a weak solution to the above stochastic heat equation was constructed in Mytnik (2002) and the pathwise uniqueness of the nonnegative solution was left as an open problem. In this paper we give an affirmative answer to this problem for certain values of alpha and beta. In particular, for alpha beta = 1 the solution to the above equation is the density of a super-Brownian motion with alpha-stable branching (see Mytnik (2002)) and our result leads to its pathwise uniqueness for 1 < alpha < root 5-1. The local Holder continuity of the solution is also obtained in this paper for fixed time t > 0.