Ergodicity of finite-energy diffusions

Wallstrom, Timothy C.

doi:10.1090/S0002-9947-1990-0986032-1

Ergodicity of finite-energy diffusions
HTML articles powered by AMS MathViewer

by Timothy C. Wallstrom

Trans. Amer. Math. Soc. 318 (1990), 735-747

DOI: https://doi.org/10.1090/S0002-9947-1990-0986032-1

PDF | Request permission

Abstract:

Recently, the existence of a class of diffusion processes with highly singular drift coefficients has been established under the assumption of "finite energy." The drift singularities of these diffusions greatly complicate their ergodicity properties; indeed, they can render the diffusion nonergodic. In this paper, a method is given for estimating the relaxation time of a finite-energy diffusion, when it is ergodic. These results are applied to show that the set of $\operatorname {spin} - \tfrac {1} {2}$ diffusions of stochastic mechanics is uniformly ergodic.

References

Philippe Blanchard and Simon Golin, Diffusion processes with singular drift fields, Comm. Math. Phys. 109 (1987), no. 3, 421–435. MR 882809
Fritz Bopp and Rudolf Haag, Über die Möglichkeit von Spinmodellen, Z. Naturforschung 5a (1950), 644–653 (German). MR 0043309
Eric A. Carlen, Conservative diffusions, Comm. Math. Phys. 94 (1984), no. 3, 293–315. MR 763381
Thaddeus George Dankel Jr., Mechanics on manifolds and the incorporation of spin into Nelson’s stochastic mechanics, Arch. Rational Mech. Anal. 37 (1970), 192–221. MR 263375, DOI 10.1007/BF00281477
R. L. Dobrušin, Central limit theorem for nonstationary Markov chains. II, Teor. Veroyatnost. i Primenen. 1 (1956), 365–425 (Russian, with English summary). MR 0097112
R. L. Dobrušin, Central limit theorem for nonstationary Markov chains. II, Teor. Veroyatnost. i Primenen. 1 (1956), 365–425 (Russian, with English summary). MR 0097112
William G. Faris, Spin correlation in stochastic mechanics, Found. Phys. 12 (1982), no. 1, 1–26. MR 659601, DOI 10.1007/BF00726872
J. Hajnal, Weak ergodicity in non-homogeneous Markov chains, Proc. Cambridge Philos. Soc. 54 (1958), 233–246. MR 96306
Marius Iosifescu, Finite Markov processes and their applications, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Ltd., Chichester; Editura Tehnică, Bucharest, 1980. MR 587116

Stochastic integrals

Derivation of the Schrödinger equation from Newtonian mechanics

150

Edward Nelson, Dynamical theories of Brownian motion, Princeton University Press, Princeton, N.J., 1967. MR 0214150
Edward Nelson, Critical diffusions, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 1–11. MR 889464, DOI 10.1007/BFb0075835

Quantum fluctuations

E. Seneta, Coefficients of ergodicity: structure and applications, Adv. in Appl. Probab. 11 (1979), no. 3, 576–590. MR 533060, DOI 10.2307/1426955
Timothy C. Wallstrom, A finite-energy bound on the approach of a diffusion to the zeros of its density, Proc. Amer. Math. Soc. 108 (1990), no. 3, 839–843. MR 1004425, DOI 10.1090/S0002-9939-1990-1004425-9
Timothy C. Wallstrom, The stochastic mechanics of the Pauli equation, Trans. Amer. Math. Soc. 318 (1990), no. 2, 749–762. MR 986033, DOI 10.1090/S0002-9947-1990-0986033-3
W. A. Zheng, Tightness results for laws of diffusion processes application to stochastic mechanics, Ann. Inst. H. Poincaré Probab. Statist. 21 (1985), no. 2, 103–124 (English, with French summary). MR 798890