Real-Valued Semigroups and (Causal) Diffusion

It can be shown that a process modeled by a strongly continuous real-valued semigroup (that has a space convolution operator as infinitesimal generator) cannot satisfy causality. By causality we mean that a characteristic feature of a process like an interface or a front must propagate with a finite speed. We present and discuss a causal model of diffusion that satisfies the semigroup property at a discrete set of time instants M:={mτ|m is an element of N0} and that in contrast to the classical diffusion model is 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 instants. It is this property of (causal) diffusion that forbids the classical limit procedure τ→0 that leads to the noncausal diffusion model in Stochastics. Finally, we give two explanations why in some cases the discretization of the noncausal diffusion model can be considered as an approximation of the causal diffusion model. In particular, we present an inhomogeneous wave equation with a time dependent coefficient that is satisfied by causal diffusion.