Abstract
It is shown that the solutions to quite general problems in nonequilibrium statistical physics and engineering can be expressed as Wiener integrals. A new way is also given for numerically evaluating these Wiener integrals.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Bryson, Jr., and M. Frazier, Smoothing for linear and nonlinear systems,Proc. of the Optimum-System Synthesis Conference, Tech. Doc. Rep. No. ASD-TRD-63-119, Flight Control Lab., ASD AFSC Wright Patterson AFB, Ohio, February 1963.
R. H. Cameron and W. T. Martin, Evaluation of various Wiener integrals by the use of Sturm-Liouville differential equations,Bull. Am. Math. Soc. 51, 73–89 (1945).
R. H. Cameron and W. T. Martin, The orthogonal development of nonlinear functions in series of Fourier-Hermite functionals,Ann. Math. 48, 385–389 (1947).
R. H. Cameron, Nonlinear Volterra functional equations,J. d'Analyse Math. 5, 470–510 (1957).
R. H. Cameron and D. A. Storvick, A translation theorem for analytic Feynman integrals,Trans. Am. Math. Soc. 125 (1), 1–6 (Oct. 1966).
S. Chandrasekhar, Stochastic problems in physics and astronomy,Rev. Mod. Phys. 15 (1) (1943); reprinted in N. Wax, ed.,Selected Papers on Noise and Stochastic Processes (Dover Publications, New York, 1954).
M. Donsker, On function space integrals, inAnalysis in Function Space, edited by W. T. Martin and A. Segal (M.I.T. Press, Cambridge, Mass., 1964).
J. L. Doob,Stochastic Processes (John Wiley and Sons, New York, 1953).
E. B. Dynkin,Markov Processes, Vols. I and II (Academic Press, New York, 1965).
A. Erdelyi,Asymptotic Expansions, (Dover Publications, New York, 1956).
R. T. Feynman and R. Hibbs,Quantum Mechanics and Path Integrals (McGraw-Hill Book Co., New York, 1965).
H. C. Finlayson, An approximation method for Wiener integrals of certain unbounded functionals,Proc. Am. Math. Soc. 18 (5), 798–803 (Oct. 1967).
B. Friedland and I. Bernstein, Estimation of the state of a nonlinear process in the presence of nongaussian noise and disturbances,J. Franklin Inst. 281 (6), 455–480 (June 1966).
B. Friedlandet al., Stability of attitude control systems acted upon by random perturbations, NASA Contractor Report CR-731.
I. M. Gelfand and A. M. Yaglom, Function space integration,J. Math. Phys. 1, 48–69 (1960).
J. Gimibre, Reduced density matrices of quantum gases I, II, and III,J. Math. Phys. June 1965, 238–251, 252–262, 1432–1446.
I. V. Girsanov, On the transformation of a class of stochastic processes by means of an absolutely continuous change of measure (in Russian),Teok. Beroyatnost, i Primen. 1960, 314–330. (There is an English translation in the JournalTheory of Probability and its Applications.)
L. Gross, Classical analysis on a Hubert space, inAnalysis in Function Space, edited by W. T. Martin and A. Segal (M.I.T. Press, Cambridge, Mass., 1964).
P. R. Halmos,Measure Theory (D. Van Nostrand Co., Princeton, N.J. 1950).
E. Hopf, On the partial differential equationμ τ +μμ x =ημ xx ,Commun. Pure Appl. Math. 3, 201–230 (1950).
K. Ito,On Stochastic Differential Equations, Mem. Am. Math. Soc. 4 (1951).
J. H. Jeans,The Dynamical Theory of Gases (Dover Publications, New York, 1954).
M. Kac,Probability and Related Topics in Physical Sciences (Interscience Publishers, London, 1959).
M. Kac, Wiener and integration in function spaces,Bull. Am. Math. Soc. 75 (1/II) (Jan. 1966).
R. E. Kalman, A new approach to linear filtering and prediction problems,J. Basic Eng. (ASME Trans.) 82D, 35–45 (1960).
R. E. Kalman and R. S. Bucy, New results in linear filtering and prediction theory,J. Basic Eng. (ASME Trans.) 83D, 95–108 (1961).
J. G. Kirkwood, The statistical mechanical theory of transport process I,J. Chem. Phys. 14, 180 (1946).
J. G. Kirkwood,et al., The statistical mechanical theory of transport processes III,J. Chem. Phys. 17 (10), 988–994 (1949).
H. J. Kushner, Dynamical equations for nonlinear filtering,”J. Diff. Equations,3(2), 179–190(1967).
D. K. C. MacDonald,Noise and Fluctuations, An Introduction (John Wiley, and Sons) New York, 1962).
W. T. Martin, and A. Segal,Analysis in Function Space (M.I.T. Press, Cambridge, Mass., 1964).
E. Nelson,Dynamical Theories of Brownian Motion (Princeton Univ. Press, Princeton, N.J. 1967).
L. Onsager and S. Machlup, Fluctuations and irreversible processes,Phys. Rev. 91 (6), 1505–1510 (1953).
M. Pincus, Gaussian processes and Hammerstein integral equations,Trans. Am. Math. Soc,134 (2), 193–214 (1968).
M. Schilder, Some asymptotic formulas for Wiener integrals,Trans. Am. Math. Soc. 25 (1), 63–85 (Oct. 1966).
M. Schilder, A Monte Carlo method for analyzing non linear circuits,Intern. J. Control 5 (2), 131–134 (1967).
M. Schilder, Expected values of functionals with respect to the Ito distribution,Pacific J. Math. 25 (2), 371–380 (1968).
M. Schilder, The Wiener integral calculus and some of its applications to engineering and physics, Research Report RC 68-1, General Precision Systems, Inc., Little Falls, N.J., February 2, 1968.
M. Schilder, Applications of Wiener integrals to fluid dynamics, A report submitted to the Foundation for the Advancement of Graduate Study in Engineering (unpublished).
S. R. S. Varadhan, Asymptotic probabilities and differential equations,Commun. Pure Appl. Math. XIX (3), 261–281 (Aug. 1966).
A. V. Skorokhod, Stochastic equations for diffusion processes in a bounded region,Theory of Probability and Its Applications VI (3), 1961.
V. Volterra,Theory of Functionals and of Integral and Integra-Differential Equations (Dover Publications, New York, 1959).
N. Wiener,Non Linear Problems in Random Theory (M.I.T. Press, Cambridge, Mass., 1958).
N. Wiener, A. Siegel, B. Rankin, and W. T. Martin,Differential Space, Quantum Systems and Prediction (M.I.T. Press, Cambridge, Mass., 1966).
M. Wonham, Stochastic problems in optimal control, IEEE Convention Record, 1963, Part 4.
M. Wonham, Lecture notes on stochastic control, Lecture Notes 67-2, Division of Applied Mathematics, Brown University, February 1967.
Author information
Authors and Affiliations
Additional information
This is a rewritten version of Research Report RC 68-1, General Precision Systems, Inc., Little Falls, New Jersey
Rights and permissions
About this article
Cite this article
Schilder, M. New applications of Wiener integrals to engineering and physics. J Stat Phys 1, 475–516 (1969). https://doi.org/10.1007/BF01106582
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01106582