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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.

##### MSC:
 60J55 Local time and additive functionals
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##### References:
 [1] Aizenman, M.; Simon, B., Brownian motion and Harnack inequality for Schrödinger operators, Commun. pure appl. math., 35, 209-273, (1982) · Zbl 0459.60069 [2] Berthier, A.M.; Gaveau, B., Critère de convergence des fonctionnelles de Kac et application en mècanique quantique et en gèomètrie, J. funct. anal., 29, 416-424, (1978) · Zbl 0398.60076 [3] Blumenthal, R.M., Getoor, R.K., 1968. Markov Processes and Potential Theory. Academic Press, New York. · Zbl 0169.49204 [4] Boukricha, A.; Hansen, W.; Hueber, H., Continuous solutions of the generalized Schrödinger equation and perturbation of harmonic spaces, Exposition. math., 5, 97-135, (1987) · Zbl 0659.35025 [5] Chung, K.L., 1983. Properties of finite gauge with an application to local time. Essays in honor of Carl-Gustav Esseen, Uppsala, Sweden, pp. 16-23. [6] Chung, K.L.; Rao, K.M., General gauge theorem for multiplicative functionals, Trans. amer. math. soc., 306, 819-836, (1988) · Zbl 0647.60083 [7] Chung, K.L., Zhao, Z., 1995. From Brownian Motion to Schrödinger’s Equation. Springer, Amsterdam. [8] Dellacherie, C., Meyer, P.A., 1982. Probabilities and Potential B. North-Holland, Amsterdam. [9] Doleans-Dade, C., Quelques applications de la formule de changement de variables pour LES semimartingales, Z. wahrscheinlichkeitstheorie verw. geb., 16, 181-194, (1970) · Zbl 0194.49104 [10] Dynkin, E.B., 1961a. Transformations of Markov Processes connected with additive functionals. Proc. Fourth Berkeley Symposium on Mathematics Statistics and Probability, Vol. 2. Univ. of California Press, pp. 117-142. [11] Dynkin, E.B., 1961b. Die Grundlagen der Theorie der Markoffschen Prozesse, Springer, Berlin (in German). [12] Dynkin, E.B., 1965. Markov Processes, Vol. I. Springer, Berlin. · Zbl 0132.37901 [13] Getoor, R.K., 1975. Markov Processes: Ray Processes and Right Processes. Lecture Notes in Mathematics, Vol. 440. Springer, Berlin. · Zbl 0299.60051 [14] Getoor, R.K., 1990. Excessive Measures. Birkhäuser, Boston. [15] Getoor, R.K., Measures not charging semipolars and equations of Schrödinger type, Potential anal., 4, 79-100, (1995) · Zbl 0816.60077 [16] Gradshteyn, I.S., Ryzhik, I.M., 1980. Table of Integrals, Series and Products. Academic Press, New York. · Zbl 0521.33001 [17] Heuser, H., 1982. Functional Analysis. Wiley, Chichester. · Zbl 0465.47001 [18] Khas’minskii, R.Z., On positive solutions of the equation au + vu = 0, Theory probab. appl., 4, 309-318, (1959) · Zbl 0089.34501 [19] Knopp, K., 1971. Theory and Applications of Infinite Series, 2nd Edition, Hafner Publishing, New York. [20] Meyer, P.A.; Walsh, J.B., Quelques applications des résolvantes de ray, Invent. math., 14, 143-166, (1971) · Zbl 0224.60037 [21] Portenko, N.I., Diffusion processes with unbounded drift coefficient, Theoret. probab. appl., 20, 27-37, (1975) · Zbl 0335.60050 [22] Sharpe, M., 1988. General Theory of Markov Processes. Academic Press, San Diego. · Zbl 0649.60079 [23] Stummer, W., 1990. The Novikov and entropy conditions of diffusion processes with singular drift. Ph.D. Thesis, University of Zurich. · Zbl 0766.60073 [24] Stummer, W., The Novikov and entropy conditions of multidimensional diffusion processes with singular drift, Probab. theory related fields, 97, 515-542, (1993) · Zbl 0794.60055 [25] Sturm, K.-Th., On the dirichlet – poisson problem for Schrödinger operators, C. R. math. rep. acad. sci. Canada, 9, 149-154, (1987) · Zbl 0648.35024 [26] Sturm, K.-Th., 1989. Störung von Hunt-Prozessen durch signierte additive Funktionale. Ph.D. Thesis, University of Erlangen/Nürnberg. [27] Voigt, J., Absorption semigroups, their generators, and Schrödinger semigroups, J. funct. anal., 67, 167-205, (1986) · Zbl 0628.47027 [28] Ying, J., Dirichlet forms perturbed by additive functionals of extended Kato class, Osaka J. math., 34, 933-952, (1997) · Zbl 0903.60062
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