  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.
