This is a preview of subscription content, log in via an institution to check access.
References
A. A. Beilinson, On solution of probabilistic non-equilibrium problems for distributed systems,Theory. Prob. Appl. 9 (1964), 469–472.
F. J. Beutler, Multivariate wide-sense process and prediction theory,Ann. Math. Stat. 34 (1964), 424–438.
J. L. Doob,Stochastic Processes, John Wiley, New York, 1953.
J. L. Doob, The elementary Gaussian processes,Ann. Math. Stat. 15 (1944), 229–282.
C. L. Dolph andM. A. Woodbury, On the relation between Green's functions and covariances of certain stochastic processes and its application to unbiased linear prediction,Trans. Amer. Math. Soc. 72 (1952), 519–550.
N. Dunford andJ. T. Schwartz,Linear Operators. I, Interscience, New York, 1958.
H. Dym, Stationary measures for the flow of a linear differential equation driven by white noise,Trans. Amer. Math. Soc. 123 (1966), 130–164.
P. L. Falb, Infinite-dimensional filtering; the Kalman-Bucy filter in Hilbert space,Information and Control 11 (1967), 102–137.
M. R. Hestenes, Relative self-adjoint operators in Hilbert space,Pacific J. Math 11 (1961), 1315–1357.
T. Hida, Canonical representations of Gaussian processes and their applications,Mem. Coll. Sci. Kyoto A33 (1960), 109–155.
E. Hille andR. S. Phillips,Functional Analysis and Semi-Groups, Amer. Math. Soc. Coll. Publ., vol. 16, Amer. Math. Soc., Providence, 1957.
G. Kallianpur andV. Mandrekar, Multiplicity and representation theory of purely non-deterministic stochastic processes,Theory Prob. Appl. 10 (1965), 553–581.
N. P. Kandelskii andN. N. Vakhaniya, A stochastic integral for operator-valued functions,Theory Prob. Appl. 12 (1967), 525–528.
V. Mandrekar, On multivariate wide-sense Markov processes,Nagoya Math. J. 33 (1968), 7–19.
V. Mandrekar andH. Salehi, The square-integrability of operator-valued functions with respect to a non-negative operator-valued measure and Kolmogorov isomorphism theorem,J. Math. Mech. (to appear).
V. Mandrekar andH. Salehi, Subordination of infinite-dimensional stationary stochastic processes,Ann. Inst. Henri Poincaré (to appear).
J. L. Masseira andJ. J. Schäffer,Linear Differential Equations and Function Spaces, Academic Press, New York, 1966.
R. Payen, Fonctions aléatoires du second ordre à valeurs dans un espace de Hilbert,Ann. Inst. Henri Poincaré 3 (1967), 323–396.
N. N. Vakhaniya, A probabilistic problem for the one-dimensional heat equation,Theory Prob. Appl. 12 (1967), 666–667.
K. Yosida,Functional Analysis, Springer-Verlag, Berlin-Heidelberg-New York, 1965.
Author information
Authors and Affiliations
Additional information
This research was supported by National Science Foundation grant GP-11626.
Rights and permissions
About this article
Cite this article
Mandrekar, V., Salehi, H. Operator-Valued wide-sense Markov processes and solutions of infinite-dimensional linear differential systems driven by white noise. Math. Systems Theory 4, 340–356 (1970). https://doi.org/10.1007/BF01704078
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01704078