Abstract
In this paper, we study questions related to the theory of stochastic processes on Lie nilpotent groups. In particular, we consider the stochastic process on the Heisenberg group H3(ℝ) whose trajectories satisfy the horizontal conditions in the stochastic sense relative to the standard contact structure on H3 (ℝ). It is shown that this process is a homogeneous Markov process relative to the Heisenberg group operation. There was found a representation in the form of a Wiener integral for a one-parameter linear semigroup of operators for which the Heisenberg sublaplacian generated by basis vector fields of the corresponding Lie algebra L(H3) is producing. The main method of solving the problem in this paper is using the path integrals technique, which indicates the common direction of further development of the results.
Similar content being viewed by others
References
D. Applebaum and S. Cohen, “Lévy processes, pseudo-differential operators and Dirichlet forms in the Heisenberg group,” Ann. Fac. Sci. Toulouse: Math., Sér. 6, 13, No. 2, 149–177 (2004).
V. I. Arnold, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1989).
V. N. Berestovskii, “Geodesics of nonholonomic left-invariant intrinsic metrics on the Heisenberg group and isoperimetric curves on the Minkowski plane,” Sib. Math. J., 35, No. 1, 1–8 (1994).
V. I. Bogachev and O. G. Smolyanov, Real and Functional Analysis: A University Course [in Russian], Regular and Chaotic Dynamics, Moscow (2009).
R. H. Cameron and W. T. Martin, “Transformations of Wiener integrals under translations,” Ann. Math., 45, No. 2, 386–396 (1944).
A. Dasgupta, S. Molahajloo, and M.-W.Wong, “The spectrum of the sub-Laplacian on the Heisenberg group,” Tˆohoku Math. J., 63, 269–276 (2011).
J. Delporte, “Fonctions aléatoires presque sûrement continues sur un intervalle fermé,” Ann. Inst. Henri Poincaré, 1, No. 2, 111–215 (1964).
V. V. Dontsov, “The systoles of uniform lattices on a three-dimensional Heisenberg group with a Carnot–Carathéodory metric,” Fundam. Prikl. Mat., 6, No. 2, 401–432 (2000).
V. V. Dontsov, “Systoles on Heisenberg groups with Carnot–Carathéodory metrics,” Sb. Math., 192, No. 3, 347–374 (2001).
B. Driver and T. Melcher, “Hypoelliptic heat kernel inequalities on the Heisenberg group,” J. Funct. Anal., 221, 340–365 (2005).
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw Hill (1965).
B. Gaveau, “Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents,” Acta Math., 139, No. 1-2, 95–153 (1977).
I. M. Gel’fand and A. M. Yaglom, “Integration in function spaces and its application to quantum physics,” Usp. Mat. Nauk, 11, No. 1 (67), 77–114 (1956).
A. V. Greshnov, “Metrics and tangent cones of uniformly regular Carnot–Carathéodory spaces,” Sib. Math. J., 47, No. 2, 209–238 (2006).
A. V. Greshnov, “Differentiability of horizontal curves in Carnot–Carathéodory quasispaces,” Sib. Math. J., 49, No. 1, 53–68 (2008).
M. Gromov, “Carnot–Carathéodory spaces seen from within,” in: Sub-Riemannian Geometry, Birkhäuser, Basel (1996).
A. Hulanicki, “The distribution of energy in the Brownian motion in the Gaussian field and analytic-hypoellipticity of certain subelliptic operators on the Heisenberg group,” Stud. Math., 56, No. 2, 165–173 (1976).
V. V. Kisil, “On the algebra of pseudodifferential operators which is generated by convolutions on the Heisenberg group,” Sib. Math. J., 34, No. 6, 1066–1075 (1993).
A. N. Kolmogorov, Foundations of the Theory of Probability, Chelsea, New York (1956).
I. M. Koval’chik, “The Wiener integral,” Russ. Math. Surv., 18, No. 1, 97–134 (1963).
S. G. Krantz, Explorations in Harmonic Analysis with Applications to Complex Function Theory and the Heisenberg Group, Springer, Berlin (2007).
P. Lévy, Processus stochastiques et mouvement brownien [in French], Paris (1965).
P. Lévy, “Le mouvement brownien plan,” Am. J. Math., 62, 487–550 (1940).
W. Magnus and A. Karrass, Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, Wiley & Sons, New York (1966).
D. Neuenschwander, Probabilities on the Heisenberg Group. Limit Theorems and Brownian Motion, Lect. Notes Math., Vol. 163, Springer, Berlin (1996).
P. K. Rashevsky, “Any two points of a totally nonholonomic space may be connected by an admissible line,” Uch. Zap. Ped. Inst. Liebknechta, Ser. Phys. Math., 3, No. 2, 83–94 (1938).
E. F. Sachkova, “Sub-Riemannian balls on the Heisenberg groups: an invariant volume,” in: J. Math. Sci., 199, No. 5, 583–587 (2014).
G. E. Shilov, “Integration in infinite-dimensional spaces and the Wiener integral,” Usp. Mat. Nauk, 18, No. 2 (110), 99–120 (1963).
E. M. Stein, Harmonic Analysis. Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press (1993).
M. E. Taylor, Noncommutative Microlocal Analysis, Pt. I, Memoirs Amer. Math. Soc., Vol. 313, Amer. Math. Soc. (1984).
S. Watanabe, “Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels,” Ann. Probab., 15, No. 1, 1–39 (1987).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 21, No. 4, pp. 67–98, 2016.
Rights and permissions
About this article
Cite this article
Mamon, S.V. The Wiener Measure on the Heisenberg Group and Parabolic Equations. J Math Sci 245, 155–177 (2020). https://doi.org/10.1007/s10958-020-04684-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-020-04684-6