On the causality of real-valued semigroups and diffusion

In this paper we show that a process modeled by a strongly continuous real-valued semigroup (that has a space convolution operator as infinitesimal generator) cannot satisfy causality. We present and analyze a causal model of diffusion that satisfies the semigroup property at a discrete set of time points $M:=\{\tau_m\,|\,m\in\N_0\}$ and that is in contrast to the classical diffusion model not smooth. More precisely, if $v$ denotes the concentration of a substance diffusing with constant speed, then $v$ is continuous but its time derivative is discontinuous at the discrete set $M$ of time points. It is this property of diffusion that forbids the classical limit procedure that leads to the noncausal diffusion model in Stochastics. Furthermore, we show that diffusion with constant speed satisfies an inhomogeneous wave equation with a time dependent coefficient.