×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.