# Real-Valued Semigroups and (Causal) Diffusion

## Abstract

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:={l_brace}m{tau}|m is an element of N{sub 0}{r_brace} 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 {tau}{yields}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.

- Authors:

- Department of Mathematics, University of Innsbruck, Technikerstrasse 21a/2, A-6020, Innsbruck (Austria)

- Publication Date:

- OSTI Identifier:
- 21611616

- Resource Type:
- Journal Article

- Journal Name:
- AIP Conference Proceedings

- Additional Journal Information:
- Journal Volume: 1389; Journal Issue: 1; Conference: ICNAAM 2011: Conference on numerical analysis and applied mathematics, Halkidiki (Greece), 19-25 Sep 2011; Other Information: DOI: 10.1063/1.3636876; (c) 2011 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0094-243X

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; CAUSALITY; DIFFUSION; IMAGE PROCESSING; MATHEMATICAL OPERATORS; STOCHASTIC PROCESSES; TIME DEPENDENCE; VELOCITY; WAVE EQUATIONS; CALCULATION METHODS; DIFFERENTIAL EQUATIONS; EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; PROCESSING

### Citation Formats

```
Kowar, Richard.
```*Real-Valued Semigroups and (Causal) Diffusion*. United States: N. p., 2011.
Web. doi:10.1063/1.3636876.

```
Kowar, Richard.
```*Real-Valued Semigroups and (Causal) Diffusion*. United States. doi:10.1063/1.3636876.

```
Kowar, Richard. Thu .
"Real-Valued Semigroups and (Causal) Diffusion". United States. doi:10.1063/1.3636876.
```

```
@article{osti_21611616,
```

title = {Real-Valued Semigroups and (Causal) Diffusion},

author = {Kowar, Richard},

abstractNote = {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:={l_brace}m{tau}|m is an element of N{sub 0}{r_brace} 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 {tau}{yields}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.},

doi = {10.1063/1.3636876},

journal = {AIP Conference Proceedings},

issn = {0094-243X},

number = 1,

volume = 1389,

place = {United States},

year = {2011},

month = {9}

}