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Random motion of strings and related stochastic evolution equations*

Published online by Cambridge University Press:  22 January 2016

Tadahisa Funaki*
Affiliation:
Department of Mathematics, Faculty of Science Nagoya University Chikusa-ku, Nagoya 464,Japan
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In this paper, we shall investigate the random motion of an elastic string by using the theory of infinite dimensional stochastic differential equations. The paper consists of three main parts and appendices. In the first part (§2), we shall derive a basic equation which describes the random motion of a string. Several properties of this equation will be investigated in § 3, 4 and 5. In the third part (§ 6), we shall deal with a stochastic differential equation on a Hilbert space as a generalization of the equation of the string.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

Footnotes

*

Supported in part by the SAKKOKAI FOUNDATION.

References

[ 1 ] Agmon, S., Lectures on elliptic boundary value problems, Princeton, N.J.: D. Van Nostrand Co., Inc., 1965.Google Scholar
[ 2 ] Arima, R., On general boundary value problem for parabolic equations, J. Math. Kyoto Univ., 4 (1964), 207243.Google Scholar
[ 3 ] Blagovescenskii, Yu. N. and Freidlin, M. I., Certain properties of diffusion processes depending on a parameter, Soviet Math. Dokl., 2 (1961), 633636.Google Scholar
[ 4 ] Cabana, E., The vibrating string forced by white noise, Z. Wahrsch. Verw. Gebiete, 15 (1970), 111130.Google Scholar
[ 5 ] Dawson, D. A., Stochastic evolution equations, Math. Biosci., 15 (1972), 287316.CrossRefGoogle Scholar
[ 6 ] Cabana, E., Stochastic evolution equations and related measure processes, J. Multivariate Anal., 5 (1975), 155.Google Scholar
[ 7 ] Doss, H. et Royer, G., Processus de diffusion associe aux mesures de Gibbs sur Rzd , Z. Wahrsch. Verw. Gebiete, 46 (1978), 107124.Google Scholar
[ 8 ] Gihman, I. I. and Skorohod, A. V., Stochastic differential equation, Springer-Verlag, Berlin Heidelberg New York, 1972.Google Scholar
[ 9 ] Holley, R. A. and Stroock, D. W., Generalized Ornstein-Uhlenbeck processes and infinite particle branching Brownian motions, Publ. RIMS Kyoto Univ., 14 (1978), 741788.Google Scholar
[10] Itô, K., Stochastic analysis in infinite dimensions, Stochastic Analysis (ed. by Friedman, A. and Pinsky, M.), Academic Press (1978), 187197.Google Scholar
[11] Kolmogorov, A. N., Zur Umkerbarkeit der statistischen Naturgesetze, Math. Ann., 113 (1937), 766772.Google Scholar
[12] Kuo, H.-H., Stochastic integrals in abstract Wiener space (II) : regularity properties, Nagoya Math. J., 50 (1973), 89116.Google Scholar
[13] Lang, R., Unendlich-dimensionale Wienerprozesse mit Wechselwirkung I, II, Z. Wahrsch. Verw. Gebiete, 38 (1977), 5572 and 39 (1977), 277299.Google Scholar
[14] Lions, J. L. and Magenes, E., Non-homogeneous boundary value problems and applications I, II, Springer-Verlag Berlin Heidelberg New York, 1972.Google Scholar
[15] Marcus, R., Parabolic Itô equations, Trans. Amer. Math. Soc, 198 (1974), 177190.Google Scholar
[16] McKean, H. P. Jr., Stochastic integrals, Academic Press, New York London, 1969.Google Scholar
[17] Molchanov, S. A., Diffusion processes and Riemannian geometry, Russian Math. Surveys, 30 (1975), 163.CrossRefGoogle Scholar
[18] Nelson, E., Dynamical theories of Brownian motion, Princeton, N. J., Princeton Univ. Press, 1967.Google Scholar
[19] Yor, M., Existence et unicite de diffusions a valuers dans un espace de Hilbert, Ann. Inst. H. Poincare, 10 (1974), 5588.Google Scholar