×

Kac’s moment formula and the Feynman-Kac formula for additive functionals of a Markov process. (English) Zbl 0962.60067

Summary: Mark Kac introduced a method for calculating the distribution of the integral \(A_v = \int ^T_0 v(X_t) d t\) for a function \(v\) of a Markov process \((X_t, t\geq 0)\) and a suitable random time \(T\), which yields the Feynman-Kac formula for the moment-generating function of \(A_v\). We review Kac’s method, with emphasis on an aspect often overlooked. This is Kac’s formula for moments of \(A_v\), which may be stated as follows. For any random time \(T\) such that the killed process \((X_t, 0\leq t < T)\) is Markov with substochastic semigroup \(K_t(x,dy)={\mathbf P}_x (X_t \in dy\), \(T > t)\), any nonnegative measurable function \(v\), and any initial distribution \(\lambda \), the \(n\)th moment of \(A_v\) is \({\mathbf P}_{\lambda}A^n_v =n!\lambda (GM_v)^n {\mathbf 1}\) where \(G=\int ^{\infty}_0 K_t d t\) is the Green’s operator of the killed process, \(M_v\) is the operator of multiplication by \(v\), and 1 is the function that is identically 1.

MSC:

60J55 Local time and additive functionals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aase, K., A conditional expectation formula for diffusion processes, J. Appl. Probab., 14, 626-629 (1977) · Zbl 0373.60102
[2] Aizenman, M.; Simon, B., Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math., 35, 209-273 (1982) · Zbl 0459.60069
[3] Athreya, K. B., Darling and Kac revisited, Sankya: Indian J. Statist., 48, 255-266 (1986) · Zbl 0697.60025
[4] Benveniste, A.; Jacod, J., Systèmes de Lévy des processus de Markov, Invent. Math., 21, 183-198 (1973) · Zbl 0265.60074
[5] Berthier, A.; Gaveau, B., Critère de convergence des fonctionelles de Kac et applications en mechanique et en géometrie, J. Funct. Anal., 29, 416-424 (1978) · Zbl 0398.60076
[6] Bertoin, J., On the Hilbert transform of the local times of a Lévy process, Bull. Sci. Math., 119, 2,, 147-156 (1995) · Zbl 0830.60066
[7] Bingham, N. H., Limit theorems in fluctuation theory, Adv. Appl. Probab., 5, 554-569 (1973) · Zbl 0273.60066
[8] Blumenthal, R.M., Getoor, R.K., 1968. Markov Processes and Potential Theory. Academic Press, New York.; Blumenthal, R.M., Getoor, R.K., 1968. Markov Processes and Potential Theory. Academic Press, New York. · Zbl 0169.49204
[9] Carmona, R., Regularity properties of Schrödinger and Dirichlet semigroups,, J. Funct. Anal., 33, 3,, 259-296 (1979) · Zbl 0419.60075
[10] Chung, K. L., A probabilistic approach to the equilibrium problem in potential theory, Ann. Inst. Fourier, 23, 313-322 (1973) · Zbl 0258.31012
[11] Chung, K.L., Williams, R.J., 1990. Introduction to Stochastic Integration. 2nd ed. Birkhäuser, Basel.; Chung, K.L., Williams, R.J., 1990. Introduction to Stochastic Integration. 2nd ed. Birkhäuser, Basel. · Zbl 0725.60050
[12] Ciesielski, Z.; Taylor, S. J., First passage times and sojourn density for Brownian motion in space and the exact Hausdorff measure of the sample path, Trans. Amer. Math. Soc., 103, 434-450 (1962) · Zbl 0121.13003
[13] Darling, D. A.; Kac, M., On occupation times for Markoff processes, Trans. Amer. Math. Soc., 84, 444-458 (1957) · Zbl 0078.32005
[14] Durrett, R., 1984. Brownian Motion and Martingales in Analysis. Wadsworth, Belmont, CA.; Durrett, R., 1984. Brownian Motion and Martingales in Analysis. Wadsworth, Belmont, CA. · Zbl 0554.60075
[15] Dynkin, E. B., Infinitesimal operators of Markov processes, Teor. Veroyatnost. i Primenen., 1, 38-60 (1956) · Zbl 0073.34901
[16] Dynkin, E.B., 1965. Markov Processes, I,II. Springer, Berlin, Heidelberg, New York, Toronto (translated from the Russian edition of 1962).; Dynkin, E.B., 1965. Markov Processes, I,II. Springer, Berlin, Heidelberg, New York, Toronto (translated from the Russian edition of 1962). · Zbl 0132.37901
[17] Dynkin, E.B., 1983. Local times and quantum fields. In: Huber, P., Rosenblatt, M. (Eds.), Seminar on Stochastic Processes. Birkhäuser, Basel, pp. 69-84.; Dynkin, E.B., 1983. Local times and quantum fields. In: Huber, P., Rosenblatt, M. (Eds.), Seminar on Stochastic Processes. Birkhäuser, Basel, pp. 69-84.
[18] Dynkin, E. B., Markov processes as a tool in field theory, J. Funct. Anal., 50, 167-187 (1983) · Zbl 0522.60078
[19] Dynkin, E. B., Gaussian and non-Gaussian random fields associated with Markov processes, J. Funct. Anal., 55, 344-376 (1984) · Zbl 0533.60061
[20] Dynkin, E. B., Polynomials of the occupation field and related random fields, J. Funct. Anal., 58, 20-52 (1984) · Zbl 0552.60075
[21] Feynman, R. J., Space-time approach to nonrelativistic quantum mechanics, Rev. Mod. Phys., 20, 367-387 (1948) · Zbl 1371.81126
[22] Fitzsimmons, P.J., 1991. Harmonic morphisms and the resurrection of Markov processes. In: Barlow, M., Bingham, N. (Eds.), Stochastic Analysis, London Mathematical Society Lecture Note Series, vol. 167. Cambridge University Press, Cambridge, pp. 71-90.; Fitzsimmons, P.J., 1991. Harmonic morphisms and the resurrection of Markov processes. In: Barlow, M., Bingham, N. (Eds.), Stochastic Analysis, London Mathematical Society Lecture Note Series, vol. 167. Cambridge University Press, Cambridge, pp. 71-90. · Zbl 0753.60069
[23] Fitzsimmons, P. J.; Getoor, R. K., On the distribution of the Hilbert transform of the local time of a symmetric Lévy process, Ann. Probab., 20, 3,, 1484-1497 (1992) · Zbl 0767.60071
[24] Fukushima, M.,Ōshima, Y., Takedam, M., 1994. Dirichlet forms and symmetric Markov processes, vol. 19 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin.; Fukushima, M.,Ōshima, Y., Takedam, M., 1994. Dirichlet forms and symmetric Markov processes, vol. 19 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin.
[25] Getoor, R. K.; Sharpe, M. J., Last exit decompositions and distributions, Indiana Univ. Math. J., 23, 377-404 (1973) · Zbl 0314.60055
[26] Getoor, R. K.; Sharpe, M. J., Last exit times and additive functionals, Ann. Probab., 1, 550-569 (1973) · Zbl 0324.60062
[27] Glover, J., Energy and the maximum principle for nonsymmetric Hunt processes, Theory Probab. Appl., 26, 745-757 (1981) · Zbl 0475.60058
[28] Glover, J., Representing last exit potentials as potentials of measures, Z. Wahrsch. Verw. Gebiete., 61, 17-30 (1982) · Zbl 0476.60071
[29] Iosifescu, M., 1980. Finite Markov processes and their applications. Wiley, New York, Chichester.; Iosifescu, M., 1980. Finite Markov processes and their applications. Wiley, New York, Chichester. · Zbl 0436.60001
[30] Itô, K., McKean, H.P., Jr., 1965. Diffusion Processes and their Sample Paths. Springer, Berlin.; Itô, K., McKean, H.P., Jr., 1965. Diffusion Processes and their Sample Paths. Springer, Berlin.
[31] Jeanblanc, M., Pitman, J., Yor, M., 1997. The Feynman-Kac formula and decomposition of Brownian paths. Comput. Appl. Math. 16, 27-52.; Jeanblanc, M., Pitman, J., Yor, M., 1997. The Feynman-Kac formula and decomposition of Brownian paths. Comput. Appl. Math. 16, 27-52. · Zbl 0877.60027
[32] Kac, M., On the distribution of certain Wiener functionals, Trans. Amer. Math. Soc., 65, 1-13 (1949) · Zbl 0032.03501
[33] Kac, M., 1951. On some connections between probability theory and differential and integral equations. In: Neyman, J. (Ed.), Proc. 2nd Berkeley Symp. Math. Stat. Prob., Univ. of California Press, Berkeley, CA, pp. 189-215.; Kac, M., 1951. On some connections between probability theory and differential and integral equations. In: Neyman, J. (Ed.), Proc. 2nd Berkeley Symp. Math. Stat. Prob., Univ. of California Press, Berkeley, CA, pp. 189-215.
[34] Karatzas, I., Shreve, S., 1988. Brownian Motion and Stochastic Calculus. Springer, Berlin.; Karatzas, I., Shreve, S., 1988. Brownian Motion and Stochastic Calculus. Springer, Berlin. · Zbl 0638.60065
[35] Kemeny, J. G.; Snell, J. L., Potentials for denumerable Markov chains, J. Math. Anal. and Appl., 3, 196-260 (1961) · Zbl 0105.33103
[36] Kent, J.T., 1983. The appearance of a multivariate exponential distribution in sojurn times for birth-death and diffusion processes. In: Probability, Statistics and Analysis. London Mathematical Society Lecture Notes. Cambridge University Press, Cambridge, pp. 161-179.; Kent, J.T., 1983. The appearance of a multivariate exponential distribution in sojurn times for birth-death and diffusion processes. In: Probability, Statistics and Analysis. London Mathematical Society Lecture Notes. Cambridge University Press, Cambridge, pp. 161-179.
[37] Kesten, H., The influence of Mark Kac on probability theory, Ann. Probab., 14, 1103-1128 (1986)
[38] Khas’minskii, R. Z., On positive solutions of the equationau+vu=0, Theory Probab. Appl., 4, 309-318 (1959) · Zbl 0089.34501
[39] Kingman, J. F.C., Markov transition probabilities. IV: Recurrence time distributions, Z. Wahrsch. Verw. Gebiete, 11, 9-17 (1968) · Zbl 0182.22804
[40] Longford, N.T., 1991. Classes of multivariate exponential and multivariate geometric distributions derived from Markov processes. In: Sampson, A.R., Block, H.W., Savits, T.H. (Eds.), Topics in Statistical Dependence, vol. 16 of Lecture Notes-Monograph Series, I. M. S., pp. 359-369.; Longford, N.T., 1991. Classes of multivariate exponential and multivariate geometric distributions derived from Markov processes. In: Sampson, A.R., Block, H.W., Savits, T.H. (Eds.), Topics in Statistical Dependence, vol. 16 of Lecture Notes-Monograph Series, I. M. S., pp. 359-369.
[41] Marcus, M. B.; Rosen, J., Moduli of continuity of local times of strongly symmetric Markov processes via Gaussian processes, J. Theoret. Probab., 5, 791-825 (1992) · Zbl 0761.60035
[42] Marcus, M. B.; Rosen, J., \(p\)-variation of the local times of symmetric stable processes and of Gaussian processes with stationary increments, Ann. Probab., 20, 1685-1713 (1992) · Zbl 0762.60069
[43] Marcus, M. B.; Rosen, J., Sample path properties of the local times of strongly symmetric Markov processes via Gaussian processes, Ann. Probab., 20, 1603-1684 (1992) · Zbl 0762.60068
[44] Marcus, M. B.; Rosen, J., Logarithmic averages for the local times of recurrent random walks and Lévy processes, Stoch. Proc. Appl., 59, 175-184 (1995) · Zbl 0853.60059
[45] Marcus, M. B.; Rosen, J., Gaussian chaos and sample path properties of additive functionals of symmetric Markov processes, Ann. Probab., 24, 1130-1177 (1996) · Zbl 0862.60065
[46] Meyer, P. A., Renaissance, recollements, mélanges, ralentissement de processus de Markov, Ann. Inst. Fourier Grenoble, 25, 465-497 (1975) · Zbl 0304.60041
[47] Meyer, P.A., Smythe, R.T., Walsh, J.B., 1972. Birth and death of Markov processes. Proc. 6th Berk. Symp. Math. Stat. Prob., vol. 3, pp. 295-305.; Meyer, P.A., Smythe, R.T., Walsh, J.B., 1972. Birth and death of Markov processes. Proc. 6th Berk. Symp. Math. Stat. Prob., vol. 3, pp. 295-305. · Zbl 0255.60046
[48] Nagylaki, T., The moments of stochastic integrals and the distribution of sojourn times, Proc. Nat. Acad. Sci. USA, 71, 746-749 (1974)
[49] Pinsky, R., A spectral criterion for the finiteness or infiniteness of stopped Feynman-Kac functionals of diffusion processes, Ann. Probab., 14, 1180-1187 (1986) · Zbl 0611.60072
[50] Pitman, J., 1974. An identity for stopping times of a Markov process. In Studies in Probability and Statistics, Jerusalem Academic Press, pp. 41-57.; Pitman, J., 1974. An identity for stopping times of a Markov process. In Studies in Probability and Statistics, Jerusalem Academic Press, pp. 41-57.
[51] Pitman, J., Occupation measures for Markov chains, Adv. Appl. Probab., 9, 69-86 (1977) · Zbl 0389.60053
[52] Pitman, J., 1981. Lévy systems and path decompositions. Seminar on Stochastic Processes, 1981, Birkhäuser, Boston, pp. 79-110.; Pitman, J., 1981. Lévy systems and path decompositions. Seminar on Stochastic Processes, 1981, Birkhäuser, Boston, pp. 79-110.
[53] Puri, P. S., A method for studying the integral functionals of stochastic processes with applications. II: Sojourn time distributions for Markov chains, Z. Wahrsch. Verw. Gebiete,, 23, 85-96 (1972) · Zbl 0226.60085
[54] Ray, D. B., Sojourn times of a diffusion process, Ill. J. Math., 7, 615-630 (1963) · Zbl 0118.13403
[55] Rogers, L.C.G., Williams, D., 1987. Diffusions, Markov Processes and Martingales, vol. II, Itô Calculus. Wiley, New York.; Rogers, L.C.G., Williams, D., 1987. Diffusions, Markov Processes and Martingales, vol. II, Itô Calculus. Wiley, New York. · Zbl 0627.60001
[56] Rogers, L.C.G., Williams, D., 1994. Diffusions, Markov Processes and Martingales, 2nd ed., vol. I, Foundations, Wiley, New York.; Rogers, L.C.G., Williams, D., 1994. Diffusions, Markov Processes and Martingales, 2nd ed., vol. I, Foundations, Wiley, New York. · Zbl 0826.60002
[57] Rosen, J., 1991. Second order limit laws for the local times of stable processes, Séminaire de Probabilités, XXV, Lecture Notes in Mathematics, vol. 1485. Springer, Berlin, pp. 407-424.; Rosen, J., 1991. Second order limit laws for the local times of stable processes, Séminaire de Probabilités, XXV, Lecture Notes in Mathematics, vol. 1485. Springer, Berlin, pp. 407-424. · Zbl 0758.60078
[58] Sharpe, M.J., 1988. General theory of Markov processes. Academic Press, New York, London.; Sharpe, M.J., 1988. General theory of Markov processes. Academic Press, New York, London. · Zbl 0649.60079
[59] Sharpe, M. J., Killing times for Markov processes, Z. Wahrsch. Verw. Gebiete, 58, 223-230 (1981) · Zbl 0475.60056
[60] Sheppard, P., On the Ray-Knight Markov property of local times, J. London Math. Soc., 31, 377-384 (1985) · Zbl 0535.60070
[61] Simon, B., 1979. Functional Integration and Quantum Physics, vol. 86 of Pure and Applied Mathematics. Academic Press, New York.; Simon, B., 1979. Functional Integration and Quantum Physics, vol. 86 of Pure and Applied Mathematics. Academic Press, New York. · Zbl 0434.28013
[62] Simon, B., Schrödinger semigroups, Bull. A. M. S., 7, 447-526 (1982) · Zbl 0524.35002
[63] Stroock, D.W., 1993. Probability Theory, an Analytic View. Cambridge University Press, Cambridge.; Stroock, D.W., 1993. Probability Theory, an Analytic View. Cambridge University Press, Cambridge. · Zbl 0925.60004
[64] Williams, D., On local time for Markov chains, Bull. A. M. S., 73, 432-433 (1967) · Zbl 0155.24503
[65] Williams, D., Fictitious states, coupled laws and local time, Z. Wahrsch. Verw. Gebiete, 11, 288-310 (1969) · Zbl 0181.21202
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.