论文标题
随机半群的均匀收敛
Uniform convergence of stochastic semigroups
论文作者
论文摘要
对于$ l^1 $ - 空间上的随机$ C_0 $ -Semigroups，鉴于Semigroup包含一个部分积分操作员，因此有大量的结果表现出与$ t \ to \ infty $的均衡相关的。这在运输方程和数学生物学中有很多应用。但是，到目前为止，部分积分运算符在定理中并没有发挥重要作用，这些定理产生了半群的均匀收敛，而不仅仅是强烈的收敛。 在本文中，我们证明，对于不可还原的随机半群，统一的融合实际上与部分相同，均匀地组成，均匀地含义千古。除了该Tauberian定理外，我们还表明，当且仅当它部分是部分积分并且双重半群时，我们的半群是均匀收敛的，并且满足了一定的不可约性条件。我们的证明是基于lasota和Yorke的下界定理的统一版本，我们将其与Banach晶格理论的各种技术结合使用。
For stochastic $C_0$-semigroups on $L^1$-spaces there is wealth of results that show strong convergence to an equilibrium as $t \to \infty$, given that the semigroup contains a partial integral operator. This has plenty of applications to transport equations and in mathematical biology. However, up to now partial integral operators do not play a prominent role in theorems which yield uniform convergence of the semigroup rather than only strong convergence. In this article we prove that, for irreducible stochastic semigroups, uniform convergence to an equilibrium is actually equivalent to being partially integral and uniformly mean ergodic. In addition to this Tauberian theorem, we also show that our semigroup is uniformly convergent if and only if it is partially integral and the dual semigroup satisfies a certain irreducibility condition. Our proof is based on a uniform version of a lower bound theorem of Lasota and Yorke, which we combine with various techniques from Banach lattice theory.
