Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Harnack inequalities in infinite dimensions   We consider the Harnack inequality for harmonic functions with respect to
three types of infinite dimensional operators. For the infinite dimensional
Laplacian, we show no Harnack inequality is possible. We also show that the
Harnack inequality fails for a large class of Ornstein-Uhlenbeck processes,
although functions that are harmonic with respect to these processes do satisfy
an a priori modulus of continuity. Many of these processes also have a coupling
property. The third type of operator considered is the infinite dimensional
analog of operators in H\"{o}rmander's form. In this case a Harnack inequality
does hold.

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The categories of this paper are: math-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: ao-sci
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The categories of this paper are: bayes-an
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The categories of this paper are: q-alg
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The categories of this paper are: hep-th
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The categories of this paper are: q-fin
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The categories of this paper are: q-bio
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The categories of this paper are: chem-ph
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The categories of this paper are: supr-con
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The categories of this paper are: solv-int
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The categories of this paper are: mtrl-th
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The categories of this paper are: stat
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The categories of this paper are: eess
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The categories of this paper are: adap-org
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The categories of this paper are: cs