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On exponentials of additive functionals of Markov processes. (English) Zbl 0996.60090
A celebrated theorem due to R. Khas’minskij [Theory Probab. Appl. 4, 309-318 (1960); translation from Teor. Veroyatn. Primen. 4, 332-341 (1959; Zbl 0089.34501)] states that \(\sup _{x\in D} \mathbf E_{x}\int ^\tau _0 f(X_{s}) ds < 1\) implies \(\sup _{x\in D}\mathbf E_{x}\exp (\int ^\tau _0 f(X_{s}) ds)<\infty \), where \((X,\mathbf P_{x})\) is a continuous time-homogeneous Markov process on a metric space \(G\), \(f\geq 0\) is a nonnegative function on \(G\), \(D\subseteq G\) a domain with a regular boundary \(\partial D\) and \(\tau \) the hitting time of \(\partial D\). This useful result has been strengthened several times since; the paper under review offers further generalizations.
Let \((\Omega , {\mathcal F}, {\mathcal F}^{s}_{t}, X, \mathbf P_{s,x})\) be a time-inhomogeneous right process on a Radon space \(G\), let \(L\) be a (possibly exploding) nonnegative right continuous additive functional of \(X\), satisfying \(L_{s,s}=0\). Let \(L_{s,t} = L^{\text{cont}}_{s,t} + L^{\text{dis}}_{s,t}\) be its decomposition into the continuous and the purely discontinuous part. Let \((\tau (s))_{s\geq 0}\) be a family of random times satisfying some mild hypotheses (so that e.g. the family of deterministic times \(\tau (s) = s\lor R\) for some \(R>0\) or the family of first hitting times after \(s\) of an open set \(A\) are included). Define the Stieltjes exponential \(\text{EXP}\) by \[ \text{EXP}(L_{s,t}) = \exp (L^{\text{cont}}_{s,t})\prod _{s<z\leq t} (1 + L_{s,z} - L_{s,z-}) \] and set \[ L^{[n+1]}_{s,t} = (n+1) \int _{[s,t]} L^{[n]}_{s,u-} dL_{s,u},\quad n\geq 1. \] Under the assumption \(\sup _{s,x} \mathbf E_{s,x} L^{\text{dis}}_{s,\tau (s)}<\infty \) several integrability properties of \(L\) and \(\text{EXP}(L)\) are shown to be equivalent: For example, it is proven that there exists \(n\geq 1\) such that \[ \sup _{s,x}\mathbf E_{s,x} L^{[n]}_{s,\tau (s)} < n! \] if and only if \(\sup _{s,x}\mathbf E_{s,x}\text{EXP} (L_{s,\tau (s)}) <\infty \) and if and only if \(\sup _{s,x} \mathbf E_{s,x}\text{EXP}((1+\varepsilon)L_{s,\tau (s)}) <\infty \) for some \(\varepsilon >0\); all suprema above are taken over \((s,x)\in [0,\infty ]\times G\). In the second part of the paper, related results are obtained also for the ordinary exponential.

60J55 Local time and additive functionals
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