Abstract
We study random walks in a Hilbert space H and representations using them of solutions of the Cauchy problem for differential equations whose initial conditions are numerical functions on H. We construct a finitely additive analogue of the Lebesgue measure: a nonnegative finitely additive measure λ that is defined on a minimal subset ring of an infinite-dimensional Hilbert space H containing all infinite-dimensional rectangles with absolutely converging products of the side lengths and is invariant under shifts and rotations in H. We define the Hilbert space H of equivalence classes of complex-valued functions on H that are square integrable with respect to a shift-invariant measure λ. Using averaging of the shift operator in H over random vectors in H with a distribution given by a one-parameter semigroup (with respect to convolution) of Gaussian measures on H, we define a one-parameter semigroup of contracting self-adjoint transformations on H, whose generator is called the diffusion operator. We obtain a representation of solutions of the Cauchy problem for the Schrödinger equation whose Hamiltonian is the diffusion operator.
Similar content being viewed by others
References
A. D. Wentzell and M. I. Freidlin, Fluctuations in Dynamical Systems Caused by Small Random Pertubations [in Russian], Nauka, Moscow (1979).
A. V. Skorokhod, “Products of independent random operators,” Russ. Math. Surveys, 38, 291–318 (1983).
I. Daletskii and S. V. Fomin, Measures and Differential Equations in Infinite-Dimensional Spaces [in Russian], Nauka, Moscow (1983).
V. I. Bogachev, Measure Theory [in Russian], Vol. 1, RKhD, Moscow (2006); English transl., Springer, Berlin (2007).
V. I. Bogachev, N. V. Krylov, and M. Röckner, “Elliptic and parabolic equations for measures,” Russ. Math. Surveys, 64, 973–1078 (2009).
I. D. Remizov, “Solution of a Cauchy problem for a diffusion equation in a Hilbert space by a Feynman formula,” Russ. J. Math. Phys., 19, 360–372 (2012).
H.-H. Kuo, Gaussian Measures in Banach Spaces (Lect. Notes Math., Vol. 463), Springer, Berlin (1975).
A. M. Vershik, “Does there exist a Lebesgue measure in the infinite-dimensional space?” Proc. Steklov Inst. Math., 259, 248–272 (2007).
V. I. Bogachev, Gaussian Measures [in Russian], Fizmatlit, Moscow (1997); English transl. (Math. Surv. Monogr., Vol. 62), Amer. Math. Soc., Providence, R. I. (1998).
E. B. Dynkin, Markov Processes [in Russian], Fizmatlit, Moscow (1963); English transl., Springer, Berlin (1965).
Yu. N. Orlov, V. Zh. Sakbaev, and O. G. Smolyanov, “Unbounded random operators and Feynman formulae,” Izv. Math., 80, 1131–1158 (2016).
I. D. Remizov, “Solution of the Schrödinger equation with the use of the translation operator,” Math. Notes, 100, 499–503 (2016).
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equation (Grad. Texts Math., Vol. 194), Springer, New York (2000).
L. S. Efremova and V. Zh. Sakbaev, “Notion of blowup of the solution set of differential equations and averaging of random semigroups,” Theor. Math. Phys., 185, 1582–1598 (2015).
A. Weil, L’Intégration dans les groupes topologiques et ses applications (Actual. Sci. Indust., Vol. 869), Herman, Paris (1940).
L. Accardy, P. Gibilisco, and I. V. Volovich, “Yang–Mills gauge fields as harmonic function for the Lévy Laplacians,” Russ. J. Math. Phys., 2, 235–250 (1994).
L. Accardy and O. G. Smolyanov, “Generalized Lévy Laplacians and Ces`aro means,” Dokl. Math., 79, 90–93 (2009).
R. Léandre and I. V. Volovich, “The stochastic Lévy Laplacians and Yang–Mills equation on manifolds,” Infin. Dimens. Anal. Quantum Probab. Relat. Top., 4, 161–172 (2001).
B. O. Volkov, “Quantum probability and Lévy Laplacians,” Russ. J. Math. Phys., 20, 254–256 (2013).
L. Accardi, Y. G. Lu, and I. V. Volovich, Quantum Theory and Its Stochastic Limit, Springer, Berlin (2001).
M. Ohya and I. V. Volovich, Mathematical Foundation of Quantum Information and Computation and Its Application to Nano- and Bio-Systems, Springer, Dordrecht (2011).
L. A. Borisov, Yu. N. Orlov, and V. Zh. Sakbaev, “Feynman formulas for averaging of semigroups generated by operators of Schrödinger type [in Russian],” Preprint No. 057, Keldysh Inst. of Applied Math., Moscow (2015).
V. Z. Sakbaev, O. G. Smolyanov, and N. N. Shamarov, “Non-Gaussian Lagrangian Feynman–Kac formulas,” Dokl. Math., 90, 416–418 (2014).
O. G. Smolyanov and E. T. Shavgulidze, Continual Integrals [in Russian], URSS, Moscow (2015).
I. P. Natanson, Theory of Functions of Real Variable, Ungar, New York (1955).
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by a grant from the Russian Science Foundation (Project No. 14-11-00687).
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 191, No. 3, pp. 473–502, June, 2017.
Rights and permissions
About this article
Cite this article
Sakbaev, V.Z. Averaging of random walks and shift-invariant measures on a Hilbert space. Theor Math Phys 191, 886–909 (2017). https://doi.org/10.1134/S0040577917060083
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577917060083