Cauchy problems of fractional order and stable processes

We consider the Cauchy problem (∂/∂t)^αu(t,x)=(∂/∂x)^βu(t,x) for 1≤α, β≤2. The aim of this paper is threefold. First, we show the existence and the uniqueness of the solution, and determine its structure. Second, we give a representation of the solution by the distributions of several stable processes. Third, we show the positivity of the fundamental solution for β=2. These results offer an interpretation to phenomena between the heat equation (α=1, β=2) and the wave equation (α=β=2).