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Last exit decompositions and regularity at the boundary of transition probabilities. (English) Zbl 0551.60077
The purpose of this paper is to give a probabilistic approach to studying the regularity at the boundary of the transition probabilities of certain hypoelliptic diffusions with boundary conditions. The main tools are last exit decompositions of Brownian motion, the Malliavin calculus, the theory of excursions, and the calculus of variations on Brownian excursions.

MSC:
60J60 Diffusion processes
60J50 Boundary theory for Markov processes
60H05 Stochastic integrals
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[1] Bismut, J.M.: M?canique al?atoire. Lecture Notes in Math. N?866. Berlin, Heidelberg, New York: Springer 1981
[2] Bismut, J.M.: A generalized formula of It? and some other properties of stochastic flows. Z. Wahrscheinlichkeitstheorie verw. Gebiete 55, 331-350 (1981) · Zbl 0456.60063
[3] Bismut, J.M.: Martingales, the Malliavin calculus and hypoellipticity under general H?rmander’s conditions. Z. Wahrscheinlichkeitstheorie verw. Gebiete 56, 469-505 (1981) · Zbl 0445.60049
[4] Bismut, J.M.: Calcul des variations stochastiques et processus de sauts. Z. Wahrscheinlichkeitstheorie verw. Gebiete 63, 147-235 (1983) · Zbl 0494.60082
[5] Bismut, J.M.: An introduction to the stochastic calculus of variations. In: Stochastic differential systems. M. Kohlmann and N. Christopeit ed. Lecture Notes in Control and Inf. Sciences n?43, pp. 33-72. Berlin, Heidelberg, New York: Springer 1982
[6] Bismut, J.M.: The calculus of boundary processes. To appear in Annales E.N.S. (1984) · Zbl 0561.60081
[7] Bismut, J.M.: Jump processes and boundary processes. Proceedings of the Katata Conference in Probability (1982). pp. 53-104, K. It? ed. Amsterdam: North-Holland 1984
[8] Bismut, J.M.: Transformations diff?rentiables du mouvement Brownien. Proceedings of the Conference in Honor of L. Schwartz (1983). To appear in Ast?risque. (1985)
[9] Bismut, J.M., Michel, D.: Diffusions conditionnelles. J. Funct. Anal. Part I: 44, 174-211 (1981), Part II: 45, 274-292 (1982) · Zbl 0475.60061
[10] Dellacherie, C., Meyer, P.A.: Probabilit?s et Potentiels. Chap. I?IV. Paris: Hermann 1975. Chap. V?VIII. Paris: Hermann 1980
[11] Dynkin, E., Vanderbei, R.J.: Stochastic waves. T.A.M.S. 275, 771-779 (1983)
[12] Greenwood, P., Pitman, J.: Splitting times and fluctuations of L?vy processes. Adv. Appl. Probability 893-902 (1980) · Zbl 0443.60037
[13] H?rmander, L.: Hypoelliptic second order differential equations. Acta Math. 117, 147-171 (1967) · Zbl 0156.10701
[14] Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam: North-Holland 1981 · Zbl 0495.60005
[15] It?, K., McKean, H.P.: Diffusion processes and their sample paths. Grundlehren der Math. Wissenschaften Band 125. Berlin, Heidelberg, New York: Springer 1974 · Zbl 0837.60001
[16] Jacod, J.: Calcul stochastique et probl?me des martingales. Lecture Notes in Math. N?714. Berlin, Heidelberg, New York: Springer 1979
[17] Jeulin, T.: Semi-martingales et grossissement d’une filtration. Lecture Notes in Math. N?833. Berlin, Heidelberg, New York: Springer 1980 · Zbl 0444.60002
[18] Jeulin, T., Yor, M.: Sur les distributions de certaines fonctionnelles du mouvement Brownien. S?minaire de Probabilit?s n? XV, p. 210-226. Lecture Notes in Math. N?850. Berlin, Heidelberg, New York: Springer 1981
[19] Kaspi, H.: On the symmetric Wiener Hopf factorization for Markov additive processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 59, 179-196 (1982) · Zbl 0477.60071
[20] Kunita, H.: On the decomposition of solutions of stochastic differential equations. In: Stochastic Integrals. D. Williams ed., pp. 213-255. Lecture Notes in Math. N?851. Berlin, Heidelberg, New York: Springer 1981 · Zbl 0474.60046
[21] Kusuoka, S., Stroock, D.: Some applications of the Malliavin calculus. Proceedings of the Katata Conference in Probability (1982), pp. 271-306, K. It? ed. Amsterdam: North-Holland 1984
[22] Malliavin, P.: Stochastic calculus of variations and hypoelliptic operators. In: Proceedings of the Conference on Stochastic differential equations of Kyoto (1976). K. It? ed. pp. 155-263. Tokyo: Kinokuniya and New York: Wiley 1978
[23] Malliavin, P.: C k hypoellipticity with degeneracy. In: Stochastic Analysis, pp. 199-214. A. Friedman and M. Pinsky ed. New York: Academic Press 1978 · Zbl 0449.58022
[24] Pitman, J.: One dimensional Brownian motion and the three dimensional Bessel process. Adv. Appl. Probability 7, 511-526 (1975) · Zbl 0332.60055
[25] Prabhu, N.U.: Wiener-Hopf factorization for convolution semi-groups. Z. Wahrscheinlichkeitstheorie verw. Gebiete 23, 103-113 (1972) · Zbl 0222.60047
[26] Rogers, L.C.G.: Williams characterization of the Brownian excursion law: proof and applications. Seminaire de Probabilit? n?XV, pp. 227-250. Lecture Notes in Math. N?850. Berlin, Heidelberg, New York: Springer 1981
[27] Shigekawa, I.: Derivatives of Wiener functionals and absolute continuity of induced measures. J. Math. Kyoto Univ. 20, 263-289 (1980) · Zbl 0476.28008
[28] Silverstein, M.: Classification of coharmonic and coinvariant functions for L?vy processes. Ann. Probability 8, 539-575 (1980) · Zbl 0459.60063
[29] Stroock, D.: Diffusion processes associated with L?vy generators. Z. Wahrscheinlichkeitstheorie verw. Gebiete 32, 209-244 (1975) · Zbl 0292.60122
[30] Stroock, D.: The Malliavin calculus and its applications to second order parabolic differential equations. Math. Systems Theory. Part I: 14, 25-65 (1981). Part II: 14, 141-171 (1981) · Zbl 0474.60061
[31] Stroock, D.: The Malliavin calculus: a functional analytic approach. J. Functional Analysis 44, 212-257 (1981) · Zbl 0475.60060
[32] Stroock, D.: Some applications of stochastic calculus to partial differential equations. Ecole de probabilit?s de Saint-Flour. Lecture Notes in Math. N?976, pp. 267-382. Berlin, Heidelberg, New York: Springer 1983
[33] Stroock, D., Varadhan, S.R.S.: Diffusion processes with boundary conditions. Comm. Pure Appl. Math. 24, 147-225 (1971) · Zbl 0227.76131
[34] Treves, F.: Introduction to pseudo-differential operators. Vol. 1. New York: Plenum Press 1981
[35] Williams, D.: Path decomposition and continuity of local time for one dimensional diffusion. Proc. London Math. Soc. Ser. 3, 28, 738-768 (1974) · Zbl 0326.60093
[36] Williams, D.: Diffusions, Markov processes and martingales. Vol. 1: Foundations. New York: Wiley 1979 · Zbl 0402.60003
[37] Derridj, M.: Un probl?me aux limites pour une classe d’op?rateurs du second ordre hypoelliptiques. Ann. Inst. Fourier 21, 99-148 (1971) · Zbl 0215.45405
[38] Ben Arous, G., Kusuoka, S., Stroock, D.: The Poisson kernel for certain degenerate elliptic operators. J. Functional Analysis 56, 171-209 (1984) · Zbl 0556.35036
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