关于实值半群与扩散的因果关系
On the causality of real-valued semigroups and diffusion
摘要：
本文证明了由强连续实值半群（具有空间卷积算子作为无穷小生成元）所模拟的过程不能满足因果关系。我们提出并分析了一个因果模型的扩散，满足半群属性在一离散集的时间点$M:=\{τ_m\,|\,m\in\N_0\}$，这是在对比经典的扩散模型不光滑。更准确地说，如果$v$表示以恒定速度扩散的物质的浓度，则$v$是连续的，但其时间导数在时间点的离散集$M$处是不连续的。正是扩散的这一性质禁止了经典的极限过程，导致了随机学中的非因果扩散模型。此外，我们证明了具有恒定速度的扩散满足一个非齐次波动方程的不定常系数。
英文摘要：
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 $M:=\{τ_m\,|\,m\in\N_0\}$ 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.
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