Abstract
We discuss the problem of reconstructing the drift coefficient of a diffusion from the knowledge of the transition probabilities outside a given bounded region in ℝd,d>1. We also give an interpretation of the solution of this inverse problem in the framework of stochastic mechanics.
Similar content being viewed by others
References
S. Albeverio, Some points of interaction between stochastic analysis and quantum theory, inStochastic Differential Systems, N. Christopeit, K. Helmes, and M. Kohlmann, eds. (Springer, Berlin, 1986), pp. 1–26.
S. Albeverio, Ph. Blanchard, F. Gesztesy, and L. Streit, Quantum mechanical low energy scattering in terms of diffusion processes, inStochastic Aspects of Classical and Quantum Systems, S. Albeverio, Ph. Combe, and M. Sirugue-Collin (Springer, Berlin, 1984), pp. 207–227.
S. Albeverio, Ph. Blanchard, M. Hazewinkel, and L. Streit, eds.,Stochastic Processes in Physics and Engineering (D. Reidel, 1988).
S. Albeverio, R. Høegh-Krohn, and L. Streit, Energy forms, Hamiltonian and distorted Brownian Paths,J. Math. Phys. 18:907 (1977).
G. Alessandrini, On the identification of the leading coefficient of an elliptic equation,Boll. Un. Mat. Ital. C (6)1985:1–25.
L. Arnold,Stochastic Differential Equations: Theory and Applications (Wiley, New York, 1974).
R. Azencottet al., Géodesiques et diffusions en temps petit,Astérisque 1981:84–85.
Ph. Blanchard, Ph. Combe, and W. Zheng, Physical and mathematical aspects of stochastic mechanics,Lecture Notes in Physics, Vol. 281 (Springer, Berlin, 1987).
E. Carlen, Existence and sample path properties of the diffusion process in Nelson's stochastic mechanics, inStochastic Processes in Mathematics and Physics I, S. Albeverio, Ph. Blanchard, and L. Streit, eds. (Springer, Berlin, 1986), pp. 25–51.
G. Dohnal, On estimating the diffusion coefficient,J. Appl. Prob. 24:105–114 (1987).
J. G. B. Beumee, H. Rabitz, An application of filtering theory to parameter identification using stochastic mechanics,J. Math. Phys. 28:1787–1794 (1987).
M. H. A. Davis, Stochastic control and nonlinear filtering, Tata Institute, Bombay (1984).
A. Friedman and B. Gustaffson, Identification of the conductivity coefficient in an elliptic equation,Siam J. Math. Anal. 18:777–787 (1987).
J. Glimm and A. Jaffe,Quantum Physics, A Functional Integral Point of View (Springer, New York, 1981).
N. Ikeda and S. Watanabe,Stochastic Differential Equation and Diffusion Processes (North-Holland, 1981).
S. Kotani, One dimensional random Schrödinger operators and Herglotz function, inProceedings Taniguchi Symposium 1985, Probabilistic methods in mathematical physics (Academic Press, Boston, 1987), pp. 219–250.
W. Loges, Estimation of parameters for Hilbert space-valued partially observable stochastic processes,J. Multivariate Anal. 20:161–174 (1986).
E. Nelson,Quantum Fluctuations (Princeton University Press, 1985).
B. Øksendahl,Stochastic Differential Equations (Springer, Berlin, 1985).
J. Pöschel and E. Trubowitz,Inverse Spectral Theory (Academic Press, Boston, 1987).
L. G. Rogers and D. Williams,Diffusion, Markov Processes and Martingales, Vol. 2,Ito Calculus (Wiley, Chichester, 1987).
L. Streit, Quantum theory and stochastic processes-Some contact points, inStochastic Processes and Their Applications, K. Ito and T. Hida, eds. (Springer, Berlin, 1986), pp. 197–213.
P. C. Sabatier, ed.,Inverse Problems (Academic Press, 1987).
D. S. Schucker, Stochastic mechanics of systems with zero potential,J. Funct. Anal. 38:146 (1980).
B. Simon,Functional Integration and Quantum Physics (Academic Press, New York, 1979).
S. R. S. Varadhan,Lectures on Diffusion Problems and Partial Differential Equations (Springer-Verlag, 1980).
E. Wong and B. Hajek,Stochastic Processes in Engineering Systems (Springer, New York, 1985).
S. Albeverio, K. Yasue, J. C. Zambrini, Euclidean quantum mechanics: analytic approach, Bochum preprint, to appear inAnn. Inst. H. Poincaré (Phys. Th.) (1989).
Author information
Authors and Affiliations
Additional information
This paper is dedicated to the dear memory of Paola Calderoni.
Rights and permissions
About this article
Cite this article
Albeverio, S., Blanchard, P., Kusuoka, S. et al. An inverse problem for stochastic differential equations. J Stat Phys 57, 347–356 (1989). https://doi.org/10.1007/BF01023648
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01023648