Summary
Consider a Markov process on the real line with a specified transition density function. Certain conditions on the latter are shown to be sufficient for the almost sure existence of a local time of the sample function which is jointly continuous in the state and time variables.
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Berman, S.M.: Local times and sample function properties of stationary Gaussian processes. Trans. Amer. Math. Soc. 137, 277–299 (1969)
Berman, S.M.: Gaussian processes with stationary increments: Local times and sample function properties. Ann. Math. Statist 41, 1260–1272 (1970)
Berman, S.M.: Local nondeterminism and local times of general stochastic processes. Ann. Inst. Henri Poincaré 19, 189–207 (1983)
Cuzick, J., Du Preez, J.P.: Joint continuity of Gaussian local times. Ann. Probability 10, 810–817 (1982)
Doob, J.L.: Stochastic Processes. New York: John Wiley 1953
Ehm, W.: Sample function properties of multi-parameter stable processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 56, 195–228 (1981)
Geman, D., Horowitz, J.: Occupation Densities. Ann. Probability 8, 1–67 (1980)
Geman, D., Horowitz, J., Rosen, J.: A local time analysis of the intersections of Brownian paths in the plane. Ann. Probability 12, 86–107 (1984)
Getoor, R.K., Kesten, H.: Continuity of local times for Markov processes. Compositio Math. 24, 277–303 (1972)
Lévy, P.: Sur certains processus stochastiques homogenes. Composition Math. 7, 283–339 (1939)
Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd Edition. Heidelberg New York: Springer 1966
Pitt, L.: Local times for Gaussian vector fields. Indiana Univ. Math. J. 27, 309–330 (1978)
Siegert, A.J.F.: On the first passage time probability problem. Physical Reviews 81, 617–623 (1951)
Trotter, H.F.: A property of Brownian motion paths. Illinois J. Math. 2, 425–432 (1958)
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This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Science Foundation, Grant MCS 82-01119
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Berman, S.M. Joint continuity of the local times of Markov processes. Z. Wahrscheinlichkeitstheorie verw Gebiete 69, 37–46 (1985). https://doi.org/10.1007/BF00532584
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DOI: https://doi.org/10.1007/BF00532584