Contractibility of manifolds by means of stochastic flows

Alexandra Antoniouk, Sergiy Maksymenko

DOI: http://dx.doi.org/10.12775/TMNA.2018.022

Abstract


In the paper [Probab.\ Theory Relat.\ Fields, {\bf 100} (1994), 417-428] Xue-Mei Li has shown that the moment stability of an SDE is closely connected with the topology of the underlying manifold. In particular, she gave sufficient condition on SDE on a manifold $M$ under which the fundamental group $\pi_1 M=0$. We prove that under similar analytical conditions the manifold $M$ is contractible, that is all homotopy groups $\pi_n M$, $n\geq1$, vanish.

Keywords


Stochastic flow; $h$-Brownian motion; homotopy type; contractibility

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References


H. Airault, Subordination de processus dans le fibré tangent et formes harmoniques, C.R. Acad. Sci. Paris Sér. A-B 282 (1976), Aiii, A1311–A1314.

L. Arnold, A formula connecting sample and moment stability of linear stochastic systems, SIAM J. Appl. Math. 44 (1984), 793–802, http://dx.doi.org/10.1137/0144057.

L. Arnold, W. Kliemann and E. Oeljeklaus, Lyapunov exponents of linear stochastic systems, Lyapunov Exponents (Bremen, 1984), Lecture Notes in Math., vol. 1186, Springer, Berlin, 1986, 85–125, http://dx.doi.org/10.1007/BFb0076835.

P.H. Baxendale and D.W. Stroock, Large deviations and stochastic flows of diffeomorphisms, Probab. Theory Related Fields 80 (1988), 169–215, http://dx.doi.org/10.1007/BF00356102.

A.-M. Berthier and B. Gaveau, Critère de convergence des fonctionnelles de Kac et application en mécanique quantique et en géométrie, J. Funct. Anal. 29 (1978), 416–424, http://dx.doi.org/10.1016/0022-1236(78)90039-3.

J. Eells and K.D. Elworthy, Wiener integration on certain manifolds, Problems in Non-Linear Analysis (C.I.M.E. IV Ciclo, V arenna, 1970), Edizioni Cremonese, Rome, 1971, 67–94.

K.D. Elworthy and S. Rosenberg, Homotopy and homology vanishing theorems and the stability of stochastic flows, Geom. Funct. Anal. 6 (1996), 51–78, http://dx.doi.org/10.1007/BF02246767.

A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.

K. Itô, The Brownian motion and tensor fields on Riemannian manifold, Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag–Leffler, Djursholm, 1963, 536–539.

K. Itô, Stochastic parallel displacement, Probabilistic methods in differential equations, (Proc. Conf., Univ. Victoria, Victoria, B.C., 1974), Springer, Berlin, 1975, 1–7; Lecture Notes in Math., Vol. 451.

R.Z. Khas’minskiı̆, On the stability of the trajectory of Markov processes, J. Appl. Math. Mech. 26 (1962), 1554–1565.

F. Kozin, On relations between moment properties and almost sure Lyapunov stability for linear stochastic systems, J. Math. Anal. Appl. 10 (1965), 342–353.

F. Kozin and S. Sugimoto, Relations between sample and moment stability for linear stochastic differential equations, Proceedings of the Conference on Stochastic Differential Equations and Applications (Park City, Utah, 1976), Academic Press, New York, 145–162.

I.G. Kozubovskaja and S.M. Hrı̄sanov, The averaging method in the problem of random parametric resonance, Ukrain. Mat. Zh. 33 (1981), 444–452.

H. Kunita, On backward stochastic differential equations, Stochastics 6 (1981/82), 293–313, http://dx.doi.org/10.1080/17442508208833209.

H. Kunita, Stochastic flows and stochastic differential equations, Cambridge Stud. Adv. Math. 24, Cambridge University Press, Cambridge, 1990.

X.-M. Li, Stochastic differential equations on noncompact manifolds: moment stability and its topological consequences, Probab. Theory Related Fields 100 (1994), 417–428, http://dx.doi.org/10.1007/BF01268987.

P. Malliavin, Formules de la moyenne, calcul de perturbations et théorèmes d’annulation pour les formes harmoniques, J. Funct. Anal. 17 (1974), 274–291.

S. Taniguchi, Stochastic flows of diffeomorphisms on an open set in Rn , Stochastics Rep. 28 (1989), 301–315.

J. Vauthier, Théorèmes d’annulation et de finitude d’espaces de 1-formes harmoniques sur une variété de Riemann ouverte, Bull. Sci. Math. (2) 103 (1979), 129–177.


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