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Elliptic differential operators and diffusion processes

Published online by Cambridge University Press:  17 April 2009

Heinz Bauer
Heinz Bauer
Affiliation:
Mathematisches Institut, der Universität Erlangen-Nürnberg, Bismarckstrasse 1 1/2, 8520 Erlangen, Federal Republic of Germany.
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Abstract

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This article develops survey lectures for general mathematical audiences which the author delivered at the 27th Annual Meeting of the Australian Mathematical Society at the University of Queensland, 1983, and at the 10th Austrian Congress of Mathematicians in Innsbruck, 1981. The central theme of these lectures was the use of probabilistic methods in the study of linear elliptic-parabolic differential equations of second order.

The starting point will be an orientative discussion of the role of Brownian motion in classical potential theory. It will then be discussed that, given an elliptic-parabolic differential operator L of a certain type, there exists a uniquely determined diffusion process which is linked with L formally in the same way in which Brownian motion is linked with the Laplace operator. The fundamental results of K. Itô, D.W.Stroock and S.R.S. Varadhan will be in the centre of this part of the paper. We will then proceed to the discussion of more refined problems of the same type for differentiable manifolds. A glimpse at stochastic Riemannian geometry will then close our tour d'horizon.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 30 , Issue 2 , October 1984 , pp. 219 - 237
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Azencott, R., “Behaviour of diffusion semi-groups at infinity”, Bull. Soc. Math. France 102, (1974) 193–240.Google Scholar
[2]Bauer, H., Probability theory and elements of measure theory. (Academic Press, London-NewYork-Toronto-Sydney-San Francisco (1981).)Google Scholar
[3]Bauer, H., Harmonic spaces and associated Markov processes. In: Potential Theory, (Centro Internazionale Matematico Estivo (C.I.M.E.), Stresa 1969. Edit. Cremonese, Rome (1970)).Google Scholar
[4]Bauer, H., Harmonische Räume. (Jahrbuch Überblicke Mathematik 1981, 9–35. Bibliographisches Institut, Mannheim (1981).)Google Scholar
[5]Berg, C. and Forst, G., Potential theory on locally compact abelian groups. (Ergebnisse der Math. 87. Springer-Verlag, Berlin-Heidelberg-New York (1975).)CrossRefGoogle Scholar
[6]Chung, K.L., Lectures from Markov processes to Brownian motion. (Grundlehren d. math. Wiss. 249, Springer-Verlag, New York-Heidelberg-Berlin (1982).)CrossRefGoogle Scholar
[7]Constantinescu, C. and Cornea, A., Potential theory on harmonic spaces. (Grundlehren d. math. Wiss. 158, Springer-Verlag, Berlin-Heidelberg-New York (1972).)CrossRefGoogle Scholar
[8]Debiard, A., Gaveau, B. et Mazet, E., “Théorèmes de comparaison en géométrie riemannienne”. Publ. RIMS, Kyoto Univ. 12, (1976) 391–425.CrossRefGoogle Scholar
[9]Ichihara, K., “Curvature, geodesics and the Brownian motion on a Riemannian manifold”. Preprint, Dept. of Applied Sciences, Faculty of Engineering, Kyushu Univ.Google Scholar
[10]Ikeda, N. and Watanabe, S., Stochastic differential equations and diffusion processes. (North Holland Mathematical Library 24 North Holland Publ. Comp., Amsterdam-Oxford-New York, and Kodansha Ltd., Tokyo (1981).)CrossRefGoogle Scholar
[11]Itô, K., “On stochastic differential equations”. Mem. Amer. Math. Soc. 4 (1951).Google Scholar
[12]Johnstone, I., A probabilistic study of linear elliptic-parabolic equations of second order. (Notes on Pure Math. 12 Aust. Nat. Univ., Canberra (1979).)Google Scholar
[13]Leha, G. and Ritter, G., “On diffusion processes and their semigroups in Hilbert spaces with application to interacting stochastic systems”. Ann. of Probability (to appear).Google Scholar
[14]Malliavin, P., Géométrie differentielle stochastique. (Les Presses de l'Université de Montréal, Montréal (1978).)Google Scholar
[15]McKean, H.P., Stochastic integrals. (Academic Press, New York-London (1969).)Google Scholar
[16]Meyer, P.A., Probability and potentials. (Blaisdell Publ. Co., Waltham, Mass.-Toronto-London (1966).)Google Scholar
[17]Pinsky, M., Stochastic Riemannian geometry. In: Probabilistic analysis and related topics, Vol. I, ((ed. by Barucha-Reid, A.T.), 199–236. Academic Press, New York-San Francisco-London (1978).)Google Scholar
[18]Priouret, P., Processus de diffusion et equations differentielles stochastiques. In: Ecole d'été de probabilités de Saint Flour III – 1973. ((ed. parMeyer, P.A., Priouret, P., Spitzer, F.). Lecture Notes in Math. 390, 37–113 (1974).)Google Scholar
[19]Priouret, P., “Processus de diffusion”. Gazette des mathématiciens Soc. math. de France 4 (1975) 71–87.Google Scholar
[20]Stroock, D.W. and Varadhan, S.R.S., “Diffusion processes with continuous coefficients I, II. Comm. Pure Appl. Math. 22 (1969) 345–400 and 479–530.CrossRefGoogle Scholar
[21]Stroock, D.W. and Varadhan, S.R.S., Multidimensional diffusion processes. (Grundlehren der math. Wiss. 233, Springer-Verlag, Berlin-Heidelberg-New York (1979).)Google Scholar
[22]Walter, R., “Konvexität in riemannschen Mannigfaltigkeiten”. Jber. d. Dt. Math.-Verein. 83 (1981) 1–31.Google Scholar
[23]Williams, D., Diffusions, Markov processes, and martingales, Vol. 1 (J. Wiley & Sons, Chichester-New York-Brisbane-Toronto (1979).)Google Scholar
[24]Williams, D., “To begin at the beginning:…” In: Stochastic Integrals, (Proc. LMS Durham Symposium 1980 (ed. by Williams, D.). Lecture Notes in Math. 851, 1–55 (1981).)Google Scholar
[25]Yau, S.T., “On the heat kernel of a complete Riemannian manifold” J. Math. Pures Appl. 57, (1978) 191–201.Google Scholar