Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The asymptotic strong Feller property does not imply the e-property for
  Markov-Feller semigroups   T. Szarek [Stud. Math. 189 (2008), {\S} 4] have discussed the relationship
between two important notions concerning Markov semigroups: the asymptotic
strong Feller property and the e-property, asserting that the former property
implies the latter one. In this very short note we rectify this issue
exhibiting a simple example of a Markov-Feller semigroup enjoying the
asymptotic strong Feller property, for which the e-property is not satisfied.
(See also the comment on a connection between the asymptotic strong Feller
property and the e-property by T. Szarek, D. Worm [ETDS 32 (2012), {\S} 1]).
Additionally we give a very simple example - in comparing with the one given by
T. Szarek [Stud. Math. 189 (2008), {\S} 4] - showing that also the converse
implication does not hold.

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The categories of this paper are: physics
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The categories of this paper are: cs
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The categories of this paper are: hep-lat
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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-ex
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The categories of this paper are: supr-con
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The categories of this paper are: adap-org
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The categories of this paper are: chao-dyn
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The categories of this paper are: atom-ph
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The categories of this paper are: stat
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The categories of this paper are: comp-gas
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The categories of this paper are: plasm-ph
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The categories of this paper are: cond-mat