Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-17T00:11:11.725Z Has data issue: false hasContentIssue false
  • English
  • Français

Wiener processes on manifolds of maps

Published online by Cambridge University Press:  14 November 2011

Peter Baxendale
Peter Baxendale
Affiliation:
Department of Mathematics, King's College, Aberdeen, Scotland
Get access Rights & Permissions [Opens in a new window]

Synopsis

The solutions of stochastic differential equations are used to construct Markov processes on the Banach manifold C(S, M) of continuous maps from a compact metric space S into a smooth complete finite dimensional Riemannian manifold M. In the special case where S is a single point the construction gives a large class of diffusion processes on the manifold M, including (under certain curvature conditions) the Brownian motion process.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 87 , Issue 1-2 , 1980 , pp. 127 - 152
Copyright
Copyright © Royal Society of Edinburgh 1980

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Aronszajn, N.. La théorie des noyaux reproduisant et ses applications. Math. Proc. Cambridge Philos. Soc. 39 (1943), 133–153.CrossRefGoogle Scholar
2Baxendale, P.. Gaussian measures on function spaces. Amer. J. Math. 98 (1976), 891–952.CrossRefGoogle Scholar
3Cantor, M.. Sobolev inequalities for Riemannian bundles. Bull. Amer. Math. Soc. 80 (1974), 239–243.CrossRefGoogle Scholar
4Chen, B. Y.. Geometry of submanifolds (New York: Dekker, 1973).Google Scholar
5Clark, J. M. C.. An introduction to stochastic differential equations on manifolds. In Geometric methods in systems theory (ed. Mayne, D. Q. and Brockett, R. W.) (Amsterdam: Riedel, 1973).Google Scholar
6Debiard, A., Gaveau, B. and Mazet, E.. Temps d'arrêt des diffusions riemanniennes. C.R. Acad. Sci. Paris Sér A 278 (1974), 723–725 and 795–798.Google Scholar
7Dudley, R. M.. Sample functions of the Gaussian process. Ann. Probab. 1 (1973), 66–103.CrossRefGoogle Scholar
8Dudley, R. M., Feldman, J. and Le Cam, L.. On seminorms and probabilities and abstract Wiener spaces. Ann. of Math. 93 (1971), 390–408.Google Scholar
9Eells, J. and Elworthy, K. D.. Stochastic dynamical systems. Control theory and topics in functional analysis, Vol. III (Vienna: International Atomic Energy Agency, 1976).Google Scholar
10Elworthy, K. D.. Measures on infinite-dimensional manifolds. In Functional integration and its applications (ed. Arthurs, A.M.) (Oxford Univ. Press, 1975).Google Scholar
11Fernique, X.. Régularité de processus gaussiens. Invent. Math. 12 (1971), 304–320.Google Scholar
12Gihman, I. I. and Skorohod, A. V.. Stochastic differential equations (Berlin: Springer, 1972).Google Scholar
13Gross, L.. Abstract Wiener spaces. Proc. Fifth Berkeley Symp. in Math. Stat. and Probability (1965), Vol. 2, Part 1, 31–40.Google Scholar
14Gross, L.. Potential theory on Hilbert space. J. Funct. Anal. 1 (1967), 123–182.CrossRefGoogle Scholar
15Gross, L.. Abstract Wiener measure and infinite dimensional potential theory. In: Lectures in Modem Analysis and Applications II. Lecture Notes in Mathematics 140, 84–116 (Berlin: Springer, 1970).Google Scholar
16Itô, K.. On stochastic differential equations on a differentiable manifold 1. Nagoya Math. J. 1 (1950), 35–47.CrossRefGoogle Scholar
17Itô, K.. On stochastic differential equations on a differentiable manifold 2. Mem. Coll. Sci. Kyoto Univ. A 28 (1953), 82–85.Google Scholar
18Jørgensen, E.. The central limit problem for geodesic random walks. Z. Wahrsch. Verw. Gebiete 32 (1975). 1–64.CrossRefGoogle Scholar
19Kallianpur, G.. Abstract Wiener processes and their reproducing kernel Hilbert spaces. Z Wahrsch. Verw. Gebiete 17 (1971), 113–123.Google Scholar
20Kushner, H. J.. On the weak convergence of interpolated Markov chains to a diffusion. Ann. Prob. 2 (1974), 40–50.Google Scholar
21Mc, H. P.Kean. Stochastic integrals (New York: Academic, 1969).Google Scholar
22Molchanov, S. A.. Strong Feller property of diffusion processes on smooth manifolds. Theory Probab. Appl. 13 (1968), 471–475.CrossRefGoogle Scholar
23Nelson, E.. An existence theorem for second order parabolic equations. Trans. Amer. Math. Soc. 88 (1958), 414–429.Google Scholar
24Palais, R. et al. Seminar on the Atiyah-Singer Index Theorem. Annals of Mathematical Studies, No. 57 (Princeton: Princeton Univ. Press 1965).Google Scholar
25Parthasarathy, K. R.. Probability measures on metric spaces (New York: Academic, 1967).Google Scholar
26Priouret, P.. Processus de diffusion et equations differentielles stochastiques. Ecole d'Eté de Probabilités de Saint Flour III. Lecture Notes in Mathematics 390 (Berlin: Springer, 1974).Google Scholar
27Sato, H.. Gaussian measure on a Banach space and abstract Wiener measure. Nagoya Math. J. 36 (1969), 65–81.CrossRefGoogle Scholar
28Skorohod, A. V.. A note on Gaussian measures in Banach space. Theory Probab. Appl. 15 (1970), 519–520.Google Scholar
29Totoki, H.. A method of construction of measures on function spaces and its applications to stochastic processes. Mem. Fac. Sci. Kyushu Univ. Ser. A 15 (1961), 178–190.Google Scholar