Existence of weak solutions to stochastic heat equations driven by truncated α-stable white noises with non-Lipschitz coefficients

Abstract

We consider a class of stochastic heat equations driven by truncated α-stable white noises for α∈(1,2) with noise coefficients that are continuous but not necessarily Lipschitz continuous. We prove the existence of weak solution in probabilistic sense, taking values in two different forms under different conditions, to such an equation using a weak convergence argument on solutions to the approximating stochastic heat equations. More precisely, for α∈(1,2) there exists a measure-valued weak solution. However, for α∈(1,5/3) there exists a function-valued weak solution, and in this case we further show that for p∈(α,5/3) the uniform p-th moment in Lp-norm of the weak solution is finite, and that the weak solution is uniformly stochastic continuous in Lp sense.