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Geometric aspects of diffusions on manifolds

Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 1362)

Keywords

  • Vector Field
  • Brownian Motion
  • Riemannian Manifold
  • Heat Kernel
  • Stable Manifold

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References

  1. A. Séminaire de Probabilités XVI, 1980/81, Supplément: Géométrie Différentielle Stochastique. Lecture Notes in Maths., 921, (1981).

    Google Scholar 

  2. B. Lyapunov Exponents. Proceedings, Bremen 1984. Eds. L. Arnold & V. Wihstutz. Lecture Notes in Mathematics, 1186, (1986).

    Google Scholar 

  3. C. From Local Times to Global Geometry, Control and Physics. Ed. K.D. Elworthy, Pitman Research Notes in Mathematics Series, 150, Longman and Wiley, 1986.

    Google Scholar 

  4. Airault, H. (1976). Subordination de processus dans le fibré tangent et formes harmoniques. C.R. Acad. Sc. Paris, Sér. A, 282 (14 juin 1976), 1311–1314.

    MathSciNet  MATH  Google Scholar 

  5. Arnold, V.I. & Avez, A. (1968). Ergodic problems of classical mechanics. New York: Benjamin.

    MATH  Google Scholar 

  6. Arnold, L. (1984). A formula connecting sample and moment stability of linear stochastic systems. SIAM J. Appl. Math., 44, 793–802.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Arnold, L. & Kliemann, W. Large deviations of linear stochastic differential equations, In "Proceedings of the Fifth IFIP Working Conference on Stochastic Differential Systems, Eisenach 1986" ed. Engelbert, Lecture Notes in Control and Information Sciences. Springer-Verlag.

    Google Scholar 

  8. Azéma, J, Kaplan-Duflo, M. & Revuz, D. (1966). Récurrence fine des processus de Markov. Ann.Inst. H. Poincaré, Sect. B, II, no. 3, 185–220.

    MATH  Google Scholar 

  9. Azencott, R. (1974). Behaviour of diffusion semigroups at infinity. Bull. Soc. Math. France, 102, 193–240.

    MathSciNet  MATH  Google Scholar 

  10. Azencott, R. et al. (1981). Géodésiques et diffusions en temps petit. Séminaire de probabilités, Université de Paris VII. Astérique 84–85. Société Mathématique de France.

    Google Scholar 

  11. Ballman, W., Gromov, M. & Schroeder, V. (1985). Manifolds of non positive curvature. Boston-Basel-Stuttgart: Birkhauser.

    CrossRef  Google Scholar 

  12. Baxendale, P. (1980). Wiener processes on manifolds of maps. Proc. Royal Soc. Edinburgh, 87A, 127–152.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Baxendale, P.H. (1986). Asymptotic behaviour of stochastic flows of diffeomorphisms: two case studies. Prob. Th. Rel. Fields, 73, 51–85.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. Baxendale, P. (1985). Moment stability and large deviations for linear stochastic differential equations. In Proceedings of the Taniguchi Symposium on Probabilistic Methods in Mathematical Physics, Kyoto 1985, ed. N. Ikede. To appear.

    Google Scholar 

  15. Baxendale, P.H. (1986). The Lyapunov spectrum of a stochastic flow of diffeomorphisms. In [B] pp. 322–337.

    Google Scholar 

  16. Baxendale, P.H. (1986). Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms. Preprint: University of Aberdeen.

    Google Scholar 

  17. Baxendale, P.H. & Stroock, D.W. (1987). Large deviations and stochastic flows of diffeomorphisms. Preprint.

    Google Scholar 

  18. Bérard, P. & Besson, G. (1987). Number of bound states and estimates on some geometric invariants. Preprint: Institut Fourier, B.P. 74, 38402, St. Martin d'Heres Cedex, France.

    MATH  Google Scholar 

  19. Berthier, A.M. & Gaveau, B. (1978). Critère de convergence des fonctionnelles de Kac et applications en mécanique quantique et en géométrie. J. Funct. Anal. 29, 416–424.

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. Besse, A.-L. (1978). Manifolds all of whose geodesics are closed. Ergebruisse der Mathematik 93, Berlin, Heidelberg, New York: Springer-Verlag.

    CrossRef  MATH  Google Scholar 

  21. Bismut, J.-M. (1984). Large deviations and the Malliavin calculus. Progress in Mathematics, 45, Boston, Basel, Stuttgard: Birkhauser.

    MATH  Google Scholar 

  22. Bismut, J.-M. (1984). The Atiyah-Singer theorems: a probabilistic approach. I & II. J. Funct. Anal. 57, 56–99 & 329–348.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. Bougerol, P. (1986). Comparaison des exposants de Lyapunov des processus Markoviens multiplicatifs.

    Google Scholar 

  24. Carverhill, A.P. (1985). Flows of stochastic dynamical systems: Ergodic Theory. Stochastics, 14, 273–317.

    CrossRef  MathSciNet  MATH  Google Scholar 

  25. Carverhill, A.P. (1985). A formula for the Lyapunov numbers of a stochastic flow. Application to a perturbation theorem. Stochastics, 14, 209–226.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. Carverhill, A.P., Chappell, M. & Elworthy, K.D. (1986). Characteristic exponents for stochastic flows. In Stochastics Processes — Mathematics and Physics. Proceedings, Bielefeld 1984. Ed. S. Albeverio et al. pp. 52–72. Lecture Notes in Mathematics 1158. Springer-Verlag.

    Google Scholar 

  27. Carverhill, A.P. (1986) A non-random Lyapunov spectrum for non-linear stochastic systems. Stochastics 17, 253–287.

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. Carverhill, A.P. & Elworthy, K.D. (1986). Lyapunov exponents for a stochastic analogue of the geodesic flow. Trans. A.M.S., 295, no. 1, 85–105.

    CrossRef  MathSciNet  MATH  Google Scholar 

  29. Carverhill, A.P. & Elworthy, K.D. (1983). Flows of stochastic dynamical systems: the functional analytic approach. Z. fur Wahrscheinlichkeitstheorie 65, 245–267.

    CrossRef  MathSciNet  MATH  Google Scholar 

  30. Carverhill, A.P. (1987). The stochastic geodesic flow: nontriviality of the Lyapunov spectrum. Preprint: Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill NC 27514, U.S.A.

    Google Scholar 

  31. Chappell, M.J. (1986). Bounds for average Lyapunov exponents of gradient stochastic systems. In [B] pp. 308–321.

    Google Scholar 

  32. Chappell, M.J. (1987). Lyapunov exponents for certain stochastic flows Ph.D. Thesis. Mathematics Institute, University of Warwick, Coventry CV4 7AL, England.

    Google Scholar 

  33. Chappell, M.J. & Elworthy, K.D. (1987). Flows of Newtonian Diffusions. In Stochastic Mechanics and Stochastic Processes, ed. A. Truman. Lecture Notes in Maths. To appear.

    Google Scholar 

  34. Chavel, I. (1984). Eigenvalues in Riemannian Geometry. Academic Press.

    Google Scholar 

  35. Cheeger, J. and Ebin, D. (1975). Comparison Theorems in Riemannian Geometry. Amsterdam: North Holland.

    MATH  Google Scholar 

  36. Cheng, S.-Y. (1975). Eigenvalue comparison theorems and its geometric applications. Math. Z., 143, 289–297.

    CrossRef  MathSciNet  MATH  Google Scholar 

  37. Cheng, S.Y. & Yau, S.T. (1975). Differential equations on Riemannian Manifolds and their Geometric Applications. Comm. Pure Appl. Maths., XXVIII, 333–354.

    CrossRef  MathSciNet  MATH  Google Scholar 

  38. Colin de Verdiere, Y., (1973) Spectre du Laplacien et longueurs des géodésiques périodiques I. Compositio Math., 27, 83–106.

    MathSciNet  MATH  Google Scholar 

  39. Cycon, H., Froese, R., Kirsch, W. & Simon, B. (1987). Schrodinger Operators with applications to quantum mechanics and global geometry. Texts and Monographs in Physics. Springer-Verlag.

    Google Scholar 

  40. Darling, R.W.

    Google Scholar 

  41. Davies, E.B. & Mandouvalos, N. (1987). Heat kernel bounds on hyperbolic space and Kleinian groups. Preprint: Maths Department, Kings college, The Strand, London WC2R 2LS.

    MATH  Google Scholar 

  42. Debiard, A., Gaveau, B. & Mazet, E. (1976). Théorèmes de comparaison en géométrie riemannienne. Publ. RIMS. Kyoto Univ., 12, 391–425.

    CrossRef  MathSciNet  MATH  Google Scholar 

  43. De Rham, G. (1955). Varietes Differentiables, Paris: Herman et Cie.

    MATH  Google Scholar 

  44. Dodziuk, J. (1983). Maximum principle for parabolic inequalities and the heat flow on open manifolds. Indiana U. Math. J., 32, 703–716.

    CrossRef  MathSciNet  MATH  Google Scholar 

  45. Dodziuk, J. (1982). L2 harmonic forms on complete manifolds. In: Seminar on Differential Geometry pp. 291–302. Princeton University Press.

    Google Scholar 

  46. Donnelly, H. and Li, P (1982). Lowr bounds for the eigenvalues of Riemannian manifolds. Michigan Math. J. 29, 149–161.

    CrossRef  MathSciNet  MATH  Google Scholar 

  47. Doob, J.L. (1984). Classical Potential Theory and its Probabilistic Counterpart. Grund. der math. Wiss. 262. New York, Berlin, Heidelberg, Tokyo: Springer-Verlag.

    MATH  Google Scholar 

  48. Elworthy, K.D. & Truman, A. (1982). The diffusion equation and classical mechanics: an elementary formula. In 'stochastic Processes in Quantum Physics’ ed. S. Albeverio et al. pp. 136–146. Lecture Notes in Physics 173. Springer-Verlag.

    Google Scholar 

  49. Elworthy, K.D. & Truman, A. (1981). Classical mechanics, the diffusion (heat) equation and the Schrodinger equation on a Riemannian manifold. J. Math. Phys. 22, no. 10, 2144–2166.

    CrossRef  MathSciNet  MATH  Google Scholar 

  50. Elworthy, K.D. (1982). Stochastic Differential Equations on Manifolds. London Math. Soc. Lecture Notes in Mathematics 70, Cambridge University Press.

    Google Scholar 

  51. Elworthy, K.D. (1982). Stochastic flows and the C0 diffusion property. Stochastics 6, no. 3–4, 233–238.

    CrossRef  MathSciNet  MATH  Google Scholar 

  52. Elworthy, K.D. & Stroock, D. (1984). Large deviation theory for mean exponents of stochastic flows. Appendix to [22] above.

    Google Scholar 

  53. Elworthy, K.D. & Rosenberg, S. (1986). Generalized Bochner theorems and the spectrum of complete manifolds. Preprint: Boston University, M.A., U.S.A.

    MATH  Google Scholar 

  54. Elworthy, K.D., Ndumu, M. & Truman, A. (1986). An elementary inequality for the heat kernel on a Riemannian manifold and the classical limit of the quantum partition function. In [C], pp. 84–99.

    Google Scholar 

  55. Elworthy, K.D. (1987). Brownian motion and harmonic forms. To appear in proceedings of the workshop on stochastic analysis at Silivri, June 1986, eds. H.K. Korezlioglu and A.S. Ustunel, Lecture Notes in Maths.

    Google Scholar 

  56. Elworthy, K.D. (1987). The method of images for the heat kernel of S3. Preprint, University of Warwick.

    Google Scholar 

  57. Eskin, L.D. (1968). The heat equation and the Weierstrass transform on certain symmetric spaces. Amer. Math. Soc. Transl., 75, 239–254.

    CrossRef  MATH  Google Scholar 

  58. Friedman, A. (1974). Non-attainability of a set by a diffusion process. Trans. Amer. Math. Soc., 197, 245–271.

    CrossRef  MathSciNet  MATH  Google Scholar 

  59. Gaffney, M.P. (1954). A special Stoke's theorem for complete Riemannian manifolds. Ann. of Math., 60, 140–145.

    CrossRef  MathSciNet  MATH  Google Scholar 

  60. Gaffney, M.P. (1954). The heat equation method of Milgram and Rosenbloom for open Riemannian manifolds. Annals of Mathematics 60, no. 3. 458–466.

    CrossRef  MathSciNet  MATH  Google Scholar 

  61. Gaveau, B. (1979). Fonctions propres et non-existence absolee d'etats liés dans certains systèmes quantiques. Comm. Math. Phys. 69, 131–169.

    CrossRef  MathSciNet  MATH  Google Scholar 

  62. Gaveau, B. (1984). Estimation des fonctionelles de Kac sur une variété compacte et premièr valeur propre de Δ + f. Proc. Japan Acad., 60, Ser. A, 361–364.

    CrossRef  MathSciNet  MATH  Google Scholar 

  63. Getzler, E. (1986). A short proof of the local Atiyah-Singer Index Theorem. Topology, 25, no. 1., 111–117.

    CrossRef  MathSciNet  MATH  Google Scholar 

  64. Gikhman, I.I. & Skorohod, A.V. (1972). Stochastic Differential Equations. Berlin, Heidelberg, New York: Springer-Verlag.

    CrossRef  Google Scholar 

  65. Greene, R.E. and Wu, H. (1979). Function Theory on Manifolds which Possess a Pole. Lecture Notes in Maths., 699. Berlin, Heidelberg, New York: Springer-Verlag.

    MATH  Google Scholar 

  66. Ikeda, N. & Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. Tokyo: Kodansha. Amsterdam, New York, Oxford: North-Holland.

    MATH  Google Scholar 

  67. Ikeda, N. & Watanabe, S. (1986). Malliavin calculus of Wiener functionals and its applications. In [C], pp. 132–178.

    Google Scholar 

  68. Ito, K. (1963). The Brownian motion and tensor fields on a Riemannian manifold. Proc. Internat. Congr. Math. (Stockholm, 1962), pp. 536–539. Djursholm: Inst. Mittag-Leffler.

    Google Scholar 

  69. Kendall, W. (1987). The radial part of Brownian motion on a manifold: a semi-martingale property. Annals of Probability, 15, no. 4, 1491–1500.

    CrossRef  MathSciNet  MATH  Google Scholar 

  70. Kifer, Yu. (1986). Ergodic Theory of Random transformations. Basel: Birkhauser.

    CrossRef  MATH  Google Scholar 

  71. Kifer, Y. (1987). A note on integrability of Cr-norms of stochastic flows and applications. Preprint: Institute of Mathematics, Hebrew University, Jerusalem.

    Google Scholar 

  72. Kifer, Y. & Yomdin, Y. (1987). Volume growth and topological entropy for random transformations. Preprint: Institute of Mathematics, Hebrew University, Jerusalem.

    MATH  Google Scholar 

  73. Kobayashi, S. & Nomizu, K. (1963). Foundations of Differential Geometry. Volume 1. New York, London: John Wiley, Interscience.

    MATH  Google Scholar 

  74. Kobayashi, S. & Nomizu, K. (1969). Foundations of differential geometry, Vol. II. As vol. I, above.

    Google Scholar 

  75. Kunita, H. (1982). On backward stochastic differential equations. Stochastics, 6, 293–313.

    CrossRef  MathSciNet  MATH  Google Scholar 

  76. Kunita, H. (1984). Stochastic differential equations and stochastic flows of diffeomorphisms. In Ecole d'Eté de Probabilités de Saint-Flour XII — 1982, ed. P.L. Hennequin, pp. 143–303. Lecture Notes in Maths. 1097. Springer.

    Google Scholar 

  77. Leandre, R. (1986). Sur le theoreme d'Atiyah-Singer. Preprint: Dept. de Mathematiques, Faculté des Sciences 25030 Besacon, France.

    MATH  Google Scholar 

  78. Le Jan, Y. Equilibre statistique pour les produits de difféomorphismes aléatoires independants. Preprint: Laboratoire de Probabilités, Université, Paris 6.

    Google Scholar 

  79. Lott, J. (1987). Supersymmetric path integrals. Commun. Math. Phys., 108, 605–629.

    CrossRef  MathSciNet  MATH  Google Scholar 

  80. Malliavin, P. (1977b). Champ de Jacobi stochastiques. C.R. Acad. Sc. Paris, 285, série A, 789–792.

    MathSciNet  MATH  Google Scholar 

  81. Malliavin, M.-P. & Malliavin, P. (1974). Factorisations et lois limites de la diffusion horizontale an-dessus d'un espace Riemannien symmetrique. In Theory du Potential et Analyse Harmonique, ed. J. Faraut, Lecture Notes in Maths. 404, Springer-Verlag.

    Google Scholar 

  82. Malliavin, P. (1974). Formule de la moyenne pour les formes harmoniques. J. Funct. Anal., 17, 274–291.

    CrossRef  MathSciNet  MATH  Google Scholar 

  83. Malliavin, M.-P, & Malliavin, P. (1975). Holonomie stochastique audessus d'un espace riemannien symmétrique. C.R. Acad. Sc. Paris, 280, Série A, 793–795.

    MathSciNet  MATH  Google Scholar 

  84. Malliavin, P. (1978). Géométrie différentielle stochastique. Séminaire de Mathématiques Supérieures. Université de Montréal.

    Google Scholar 

  85. Markus, L. (1986). Global Lorentz geometry and relativistic Brownian motion. In [C], pp. 273–287.

    Google Scholar 

  86. Meyer, P.A. (1981). Géométrie différentielle stochastique (bis). In [A], pp. 165–207.

    Google Scholar 

  87. Meyer, P.A. (1981). Flot d'une equation différentielle stochastique. In Séminaire de Probabilités XV, 1979/80, eds. J. Azema and M. Yor, 103–117. Lecture Notes in Maths 860. Berlin, Heidelberg, New York: Springer-Verlag.

    Google Scholar 

  88. Meyer, P.A. (1986). Elements de probabilities quantiques In Séminaire de Probabilités XX, 1984/85, eds. J. Azema and M. Yor. Lecture Notes in Maths. 1204. Springer.

    Google Scholar 

  89. Milnor, J. (1963). Morse Theory. Annals of Math. Studies, 51, Princeton: Princeton University Press.

    CrossRef  MATH  Google Scholar 

  90. Molchanov, S.A., (1975). Diffusion processes and Riemannian geometry. Usp. Math. Nauk, 30, 3–59. English translation: Russian Math. Surveys, 30, 1–63.

    MathSciNet  MATH  Google Scholar 

  91. Ndumu, M.N. (1986). An elementary formula for the Dirichlet heat kernel on Riemannian manifolds. In [C], pp.320–328.

    Google Scholar 

  92. Ndumu, M.N. Ph.D. Thesis, Maths Dept. University of Warwick. In preparation.

    Google Scholar 

  93. Ndumu, M.N. (1987). The heat kernel of the standard 3-sphere and some eigenvalue problems. Submitted to Proc. Edinburgh Math. Soc.

    Google Scholar 

  94. Pinsky, M. (1978). Stochastic Riemannian geometry. In Probabilistic Analysis and Related Topics, 1, ed. A.T. Bharucha Reid. London, New York: Academic Press.

    Google Scholar 

  95. Pinsky, M.A. (1978). Large deviations for diffusion processes. In Stochastic Analysis, eds. Friedman and Pinsky, pp. 271–283. New York, San Francisco, London: Academic Press.

    Google Scholar 

  96. Reed, M. and Simon, B. (1978). Methods of Modern Mathematical Physics IV: Analysis of Operators. New York, San Francisco, London: Academic Press.

    MATH  Google Scholar 

  97. Ruelle, D. (1979). Ergodic Theory of Differentiable Dynamical Systems, Publications I.H.E.S., Bures-sur-Yvette, France.

    MATH  Google Scholar 

  98. Schwartz, L. (1973). Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures. Tata Institute Studies in Mathematics 6. Bombay: Oxford University Press.

    MATH  Google Scholar 

  99. Schwartz, L. (1980). Semi-martingales sur des variétés et martingales conformes sur des variétés analytiques complexes. Lecture Notes in Maths., 780, Springer-Verlag.

    Google Scholar 

  100. Schwartz, L. (1982). Géométrie différentielle du 2-eme ordre, semimartingales et equations différentielles stochastique sur une variété différentielle. In [A] pp. 1–149.

    Google Scholar 

  101. Strichartz, R.S. (1983). Analysis of the Laplacian on the complete Riemannian manifold, J. of Functional Anal. 52, 48–79.

    CrossRef  MathSciNet  MATH  Google Scholar 

  102. Strichartz, R.S. (1986). LP Contractive projections and the heat semigroup for differential forms. Jour. of Functional Anal., 65, 348–357.

    CrossRef  MathSciNet  MATH  Google Scholar 

  103. Stroock, D.W. & Varadhan, S.R.S. (1979). Multidimensional Diffusion Processes. Berlin, Heidelberg, New York: Springer-Verlag.

    MATH  Google Scholar 

  104. Sullivan, D. (1987). Related aspects of positivity in Riemannian geometry. J. Differential Geometry, 25, 327–351.

    MathSciNet  MATH  Google Scholar 

  105. Sunada, T. (1982). Trace formula and heat equation asymptotics for a non-positively curved manifold. American J. Math. 104, 795–812.

    CrossRef  MathSciNet  MATH  Google Scholar 

  106. Van den Berg, M. & Lewis, J.T. (1985). Brownian motion on a hypersurface. Bull. London Math. Soc., 17, 144–150.

    CrossRef  MathSciNet  MATH  Google Scholar 

  107. Vauthier, J. (1979). Théoremes d'annulation et de finitude d'espaces de 1-formes harmoniques sur une variété de Riemann ouverte. Bull Sc. Math., 103, 129–177.

    MathSciNet  MATH  Google Scholar 

  108. Vilms, J. (1970). Totally geodesic maps. J. Differential Geometry, 4, 73–99.

    MathSciNet  MATH  Google Scholar 

  109. Watling, K.D. (1986). Formulae for solutions to (possibly degenerate) diffusion equations exhibiting semi-classical and small time asymptotics. Ph.D. Thesis, University of Warwick.

    Google Scholar 

  110. Watling, K.D. (1987). Formulae for the heat kernel of an elliptic operator exhibiting small time asymptotics. In Stochastic Mechanics and Stochastic Processes, ed. A. Truman. Lecture Notes in Maths. To appear.

    Google Scholar 

  111. Witten, E. Supersymmetry and Morse theory, J. Diff. Geom. 17 (1982), 661–692.

    MathSciNet  MATH  Google Scholar 

  112. Yau, S.-T. (1975). Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math., 28, 201–228.

    CrossRef  MathSciNet  MATH  Google Scholar 

  113. Yau, S.-T. (1976). Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Univ. Math. J., 25, No. 7, 659–670.

    CrossRef  MathSciNet  MATH  Google Scholar 

  114. Yosida, K. (1968). Functional Analysis. (Second Edition). Grundlehren der math. Wissenschaften, 123, Berlin, Heidelberg, New York: Springer-Verlag.

    CrossRef  Google Scholar 

  115. Azencott, R. (1986) Une Approche Probabiliste du Théoréme d'Atiyah-Singer, d'après J.M. Bismut. In Séminaire Bourbaki, 1984–85. Astérisque, 133–134, 7–8. Société Mathématique de France.

    Google Scholar 

  116. Hsu, P. (1987). Brownian motion and the Atiyah-Singer index theorem. Preprint (present address: University of Illinois at Chicago).

    Google Scholar 

  117. Mañé, R. & Freire, A. (1982). On the entropy of the geodesic flow in manifolds without conjugate points. Invent. Math., 69, 375–392.

    CrossRef  MathSciNet  MATH  Google Scholar 

  118. Pesin Ya. B. (1981). Geodesic flows with hyperbolic behaviour of the trajectories and objects connected with them. Russian Math. Surveys, 36, no.4, 1–59.

    CrossRef  MathSciNet  MATH  Google Scholar 

  119. Rogers, L.C.G. & Williams, D. (1987). Diffusions, Markov processes and martingales, Vol.2: Itô calculus. Wiley series in probability and mathematical statistics. Chichester: Wiley.

    Google Scholar 

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Elworthy, D. (1988). Geometric aspects of diffusions on manifolds. In: Hennequin, PL. (eds) École d'Été de Probabilités de Saint-Flour XV–XVII, 1985–87. Lecture Notes in Mathematics, vol 1362. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086183

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