Logarithmic Sobolev inequalities and exponential entropy decay in non-commutative algebras
Logarithmic Sobolev inequalities and exponential entropy decay in non-commutative algebras
复制标题
非交换代数中的对数 Sobolev 不等式和指数熵衰减
DOI：
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发表时间：
2014
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通讯作者：
R. Carbone
中科院分区：
文献类型：
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作者：
R. Carbone
研究了von Neumann代数上马氏演化的（紧）对数Sobolev不等式、熵衰减和谱间隙不等式之间的关系。我们证明了对数Sobolev不等式（在Olkiewicz和Zegarlinski定义的非交换形式）包含谱间隙不等式，常数之间具有最佳关系。此外，我们表明，一个适当的相对熵的均匀指数衰减是等价的log-Sobolev不等式的修改版本，这种熵衰减原来是由通常的log-Sobolev不等式添加一些正则性条件的二次型所隐含的。
We study the relations between (tight) logarithmic Sobolev inequalities, entropy decay and spectral gap inequalities for Markov evolutions on von Neumann algebras. We prove that log-Sobolev inequalities (in the non-commutative form defined by Olkiewicz and Zegarlinski) imply spectral gap inequalities, with optimal relation between the constants. Furthermore, we show that a uniform exponential decay of a proper relative entropy is equivalent to a modified version of log-Sobolev inequalities; this entropy decay turns out to be implied by the usual log-Sobolev inequality adding some regularity conditions on the quadratic forms.