预解正线性算子的归约现象
Resolvent Positive Linear Operators Exhibit the Reduction Phenomenon
摘要：
Kato证明了预解式正线性算子A和乘法算子V的组合的谱界s（aA + bV）在b \in R中是凸的.这里通过一个初等引理表明，s（a A + b V）在a > 0时也是凸的，并且值得注意的是，当存在时，\partial s（a A + b V）/ \partial a <= s（A）。扩散通常具有s（A）<= 0，使得对于具有空间异质生长或衰减速率的扩散，更大的混合减少生长。当A是拉普拉斯算子或二阶椭圆算子，或非局部扩散算子时，扩散演化的模型特别发现了这个结果，这意味着选择减少扩散。这些情况在这里被证明是一个单一的、广泛的、“减少”现象的一部分。
英文摘要：
The spectral bound, s(a A + b V), of a combination of a resolvent positive linear operator A and an operator of multiplication V, was shown by Kato to be convex in b \in R. This is shown here, through an elementary lemma, to imply that s(a A + b V) is also convex in a > 0, and notably, \partial s(a A + b V) / \partial a <= s(A) when it exists. Diffusions typically have s(A) <= 0, so that for diffusions with spatially heterogeneous growth or decay rates, greater mixing reduces growth. Models of the evolution of dispersal in particular have found this result when A is a Laplacian or second-order elliptic operator, or a nonlocal diffusion operator, implying selection for reduced dispersal. These cases are shown here to be part of a single, broadly general, `reduction' phenomenon.
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