Summary
Here we discuss the regularity of solutions of SDE's and obtain conditions under which a SDE on a complete Riemannian manifoldM has a global smooth solution flow, in particular improving the usual global Lipschitz hypothesis whenM=R n. There are also results on non-explosion of diffusions.
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Bakry, D: Un critére de non-explosion pour certaines diffusions sur une varieté Riemannienne compléte. C. R. Acad. Sc. Paris, t. 303, SérieI (1) 23–26 (1986)
Baxendale, P.: Wiener processes on manifolds of maps. Proceedings of the Royal Society of Edinburgh87A 127–152 (1980)
Blagovescenskii, Y., Freidlin, M.: Some properties of diffusion processes depending on a parameter. Soviet Math.2 (DAN 138) 633–636 (1961)
Carverhill, A., Elworthy, D.: Flows of stochastic dynamical systems: the functional analytic approach. Z. Wahrescheinlichkeitstheor. Verw. Geb.65, 245–267 (1983)
Cranston, M. Kendall, W.S., March, P.: The radial part of Brownian motion II: its life and time on the cut-locus. Probab. Theory Relat. Fields96, 353–368 (1993)
Elworthy, K.D.: Geometric aspects of diffusions on manifolds. In Hennequin, P.L. (ed.) Ecole d'Eté de Probabilités de Saint-Flour XV–XVII, 1985, 1987. (Lect. Notes Math. vol. 1362, pp. 276–425) Berlin Heidelberg New York: Springer 1988
Elworthy, K.D.: Stochastic flows on Riemannian manifolds. In Pinsky, M.A., Wihstutz, V. (ed.) Diffusion processes and related problems in analysis, vol. II pp. 37–72 Birkhauser Progress in Probability. Boston: Birkhauser 1992
Elworthy, K.D.: Stochastic differential geometry. Bull. Sc. Math. 2e série117, 7–28. (1993)
Elworthy, K.D.: Stochastic dynamical systems and their flows. In Freidman, A., Pinsky, M., (ed.) (Stochatic Analysis pp. 79–95) London, New York: Academic Press 1978
Elworthy, K.D.: Stochastic Differential Equations on Manifolds., (Lect. Notes Series 70) Cambridge University Press, 1982
Elworthy, K.D.: Stochastic flows and thec 0 property. Stochastics,0, 1–6 (1982)
Elworthy, K.D., Li, X.-M.: Formulae for the derivatives of heat semigroups. Warwick (Preprints 45/1993) J. Funct. Anal. (to appear)
Greene, R.E., Wu, H.: Function theory on manifolds which possess a pole. (Lect. Notes Math. vol., 699) Berlin Heidelberg New York: Springer 1979
Krylov, N.: Controlled diffusion processes. Berlin Heidelberg New York: Springer 1980
Kunita, H.: On the decomposition of solutions of differential equations. In Williams, D. editor, Stochastic integrals. (Lect. Notes Math. vol., 851, pp. 213–255) Berlin Heidelberg New York: Springer 1980
Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, 1990
Li, X.-M.: Stochastic differential equations on noncompact manifolds: moment stability and its topological consequences. Warwick (Preprints 15/1994) Probab. Theory Relat. Fields (to appear)
Li, X.-M.: Stochastic Flows on Noncompact Manifolds. Ph.D. thesis, University of Warwick, 1992
Munkres, J.: Elementary Differential Geometry. (revised edition) Princeton University Press, 1966
Norris, J.: Simplified Malliavin calculus. In Azéma, J., Yor, M. (ed.) Seminaire de Probability XX. (Lect. Notes Math. vol., 1204, pp. 101–130) Berlin Heidelberg New York: Springer 1986
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Berlin Heidelberg New York: Springer 1991
Taniguchi, S.: Stochastic flows of diffeomorphisms on an open set ofR n Stochastics Stochastics Rep.28, 301–315 (1989)
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Research supported by SERC grant GR/H67263
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Li, XM. Strong p-completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds. Probab. Th. Rel. Fields 100, 485–511 (1994). https://doi.org/10.1007/BF01268991
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DOI: https://doi.org/10.1007/BF01268991