Joint Hölder Continuity of Parabolic Anderson Model

Abstract

We obtain the Holder continuity and joint Holder continuity in space and time for the random field solution to the parabolic Anderson equation $$(\partial_t-\frac{1}{2}\Delta)u=u\diamond\dot{W}$$ in d-dimensional space, where Ẇ is a mean zero Gaussian noise with temporal covariance γ0 and spatial covariance given by a spectral density µ(ξ). We assume that $$\gamma_0(t)\leq{c}|t|^{\alpha_0}$$ and $$|\mu(\xi)|\leq{c}\prod_{i=1}^d|\xi_i|^{-\alpha_i}\;{\rm{or}}\;|\mu(\xi)|\leq{c}|\xi|^{-\alpha}$$ , where αi, i = 1, …, d (or α) can take negative value.