PUB - Publikationen an der Universität Bielefeld

We study the large-time behaviour of the fundamental solution of parabolic equations with an elliptic part being non-local convolution-type operator. We assume that this operator is a generator of a Markov jump process, and that its convolution kernel decays at least exponentially at infinity. The fundamental solution shows rather different asymptotic behaviour depending on whether vertical bar x vertical bar less than or similar to root t, or root t << vertical bar x vertical bar << t, or vertical bar x vertical bar similar to t, or vertical bar x vertical bar >> t. In each of these regions we obtain sharp pointwise estimates for the fundamental solution.