Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Necessary and sufficient conditions for the $r$-excessive local
  martingales to be martingales   We consider the decreasing and the increasing $r$-excessive functions
$\varphi_r$ and $\psi_r$ that are associated with a one-dimensional
conservative regular continuous strong Markov process $X$ with values in an
interval with endpoints $\alpha < \beta$. We prove that the $r$-excessive local
martingale $\bigl( e^{-r (t \wedge T_\alpha)} \varphi_r (X_{t \wedge T_\alpha})
\bigr)$ $\bigl($resp., $\bigl( e^{-r (t \wedge T_\beta)} \psi_r (X_{t \wedge
T_\beta}) \bigr) \bigr)$ is a strict local martingale if the boundary point
$\alpha$ (resp., $\beta$) is inaccessible and entrance, and a martingale
otherwise.

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The categories of this paper are: nucl-ex
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The categories of this paper are: plasm-ph
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The categories of this paper are: econ
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The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
34
The categories of this paper are: chem-ph
26
The categories of this paper are: bayes-an
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The categories of this paper are: atom-ph
9
The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: stat
1
The categories of this paper are: cs
12
The categories of this paper are: nucl-th
0
The categories of this paper are: hep-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-th