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Permanental fields, loop soups and continuous additive functionals. (English) Zbl 1316.60075

In this paper, the authors define a permanental field, \(\psi=\{\psi(\nu): \nu\in \mathcal{V}\}\), a new stochastic process indexed by a space of measures \(\nu\) on a set \(S\). It is determined by a kernel \(u(x, y)\), \(x, y\in S\), that need not be symmetric and is allowed to be infinite on the diagonal. They show that these fields exist whenever \(u(x,y)\) is the potential density of a transient Markov process \(X\) in \(S\).
A permanental field \(\psi\) can be realised as the limit of a renormalized sum of continuous additive functionals determined by a loop soup of \(X\), which they construct. They also obtain a new Dynkin type isomorphism theorem that relates \(\psi\) to continuous additive functionals of \(X\) and can be used to analyze them.

MSC:

60G60 Random fields
60K99 Special processes
60J55 Local time and additive functionals
60G17 Sample path properties
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References:

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