Abstract
A class of stochastic differential equations with highly singular drift fields is considered. Using a purely probabilistic approach, we can show the unattainability of the nodal set. Moreover, a global existence and uniqueness theorem for diffusion processes with singular drift fields is established. The finite action condition of Carlen and Zheng can be modified. We relate our results to the diffusions which describe the time evolution of quantum systems in stochastic mechanics.
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Gihman, I.I., Skohorod, A.V.: Stochastic differential equations. Berlin, Heidelberg, New York: Springer 1972
Stroock, D.W., Varadhan, S.R.S.: Multidimensional diffusion processes. Berlin, Heidelberg, New York: Springer 1979
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam: North-Holland 1981
Albeverio, S., Høegh-Krohn, R.: A remark on the connection between stochastic mechanics and the heat equation. J. Math. Phys.15, 1745 (1974)
Carmona, R.: Processus de diffusion gouverné par la forme de Dirichlet de l'opérateur de Schrödinger. In: Séminaire de probabilités XIII. Dellacherie, C. et al. (eds.). Berlin, Heidelberg, New York: Springer 1979
Nagasawa, M.: Segregation of a population in an environment. J. Math. Biol.9, 213 (1980)
Albeverio, S., Fukushima, M., Karwowski, W., Streit, L.: Capacity and quantum mechanical tunneling. Commun. Math. Phys.81, 501 (1981)
Albeverio, S., Blanchard, Ph., Høegh-Krohn, R.: Newtonian diffusions and planets, with a remark on non-standard dirichlet forms and polymers. In: Stochastic analysis and applications. Truman, A., Williams, D. (eds.). Lecture Notes in Mathematics, Vol. 1095. Berlin, Heidelberg, New York: Springer 1984
Fukushima, M.: Energy forms and diffusion processes. In: Mathematics + physics, lectures on recent results. Streit, L. (ed.). New York: World Scientific 1985
Blanchard, Ph., Zheng, W.: Pathwise conservation law for stationary diffusion processes. In: Stochastic processes in classical and quantum systems. Albeverio, S. et al. (eds.). Lecture Notes in Physics. Berlin, Heidelberg, New York: Springer (to appear)
Nelson, E.: Critical diffusions. In: Séminaire de probabilités XIX. Azéma, J., Yor, M. (eds.). Berlin, Heidelberg, New York: Springer 1985
Nelson, E.: Quantum fluctuations. Princeton, NJ: Princeton University Press 1985
Carlen, E.A.: Conservative diffusions. Commun. Math. Phys.94, 293 (1984)
Zheng, W.A.: Tightness results for laws of diffusion processes. Application to stochastic mechanics. Ann. Inst. Henri Poincaré B21, 103 (1985)
Zheng, W.A., Meyer, P.A.: Sur la construction de certaines diffusions. In: Séminaire de probabilités XX. Lecture Notes in Mathematics. Berlin, Heidelberg, New York: Springer (to appear)
Guerra, F.: Carlen processes: a new class of diffusions with sigular drifts. In: Quantum probability and applications II. Accardi, L., von Waldenfels, W. (eds.). Lecture Notes in Mathematics, Vol. 1136. Berlin, Heidelberg, New York: Springer 1985
Föllmer, H.: Time reversal on Wiener space. In: Stochastic processes — mathematics and physics (1st BiBoS-Symposium). Lecture Notes in Mathematics, Vol. 1158. Berlin, Heidelberg, New York: Springer 1986
Fürth, R.: Über einige Beziehungen zwischen klassischer Statistik und Quantenmechanik. Z. Physik81, 143 (1933)
Nelson, E.: Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev.150, 1079 (1966)
Nelson, E.: Dynamical theories of Brownian motion. Princeton, NJ: Princeton University Press 1967
Guerra, F., Morato, L.M.: Quantization of dynamical systems and stochastic control theory. Phys. Rev. D27, 1774 (1983)
Reed, M., Simon, B.: Methods of modern mathematical physics. Vol. II. Fourier analysis and self-adjointness. New York: Academic Press 1975
Wilcox, C.H.: Uniform asymptotic estimates for wave packets in the quantum theory of scattering. J. Math. Phys.6, 611 (1965)
Sobolev, S.L.: Applications of functional analysis in mathematical physics, translations of mathematical monographs. Providence, RI: Am. Math. Soc. 1963
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Communicated by A. Jaffe
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Blanchard, P., Golin, S. Diffusion processes with singular drift fields. Commun.Math. Phys. 109, 421–435 (1987). https://doi.org/10.1007/BF01206145
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DOI: https://doi.org/10.1007/BF01206145