On Hardy-Littlewood Inequality for Brownian Motion on Riemannian Manifolds (original) (raw)
Related papers
Range of fluctuation of Brownian motion on a complete Riemannian manifold
The Annals of Probability, 1998
We investigate the escape rate of the Brownian motion Wx(t) on a complete noncompact Riemannian manifold. Assuming that the manifold has at most polynomial volume growth and that its Ricci curvature is bounded below, we prove that dist(Wx(t), x) ≤ Ct log t for all large t with probability 1. On the other hand, if the Ricci curvature is nonnegative and the volume growth is at least polynomial of the order n > 2, then dist(Wx(t), x) ≥ √ Ct log 1 n−2 t log log 2+ε n−2 t again for all large t with probability 1 (where ε > 0). 1991 Mathematics Subject Classification. Primary 58G32, 58G11; Secondary 60G17, 60F15. Key words and phrases. Brownian motion, heat kernel, Riemannian manifold, escape rate, the law of the iterated logarithm.
Diffusion processes on complete riemannian manifolds
Acta Mathematicae Applicatae Sinica, 1994
In this paper, a basic estimate for the conditional Riemannian Brownian motion on a complete manifold with non-negative Ricci curvature is established. Applying it to the heat kernel estimate of the operator 1A-~-b, we obtain the Aronson's estimate for the operator 1/~ _}. b, which can be regarded as an extension of Peter Li and S.T. Yau's heat kernel estimate for the Laplace-Beltraml operator.
Isoperimetric Constants, the Geometry of Ends, and Large Time Heat Diffusion in Riemannian Manifolds
Proceedings of The London Mathematical Society, 1991
Let M be a non-compact Riemannian manifold of dimension n s= 2, with associated Laplace-Beltrami operator A acting on functions on M, and with attendant minimal positive heat kernel p (x, y, t), where x,y e Af, and t >0. We are interested in those aspects of the geometry of M related to the inequalities of the type
Annales de l’institut Fourier, 2005
We study the rate of concentration of a Brownian bridge in time one around the corresponding geodesical segment on a Cartan-Hadamard manifold with pinched negative sectional curvature, when the distance between the two extremities tends to infinity. This improves on previous results by A. Eberle [7], and one of us [21]. Along the way, we derive a new asymptotic estimate for the logarithmic derivative of the heat kernel on such manifolds, in bounded time and with one space parameter tending to infinity, which can be viewed as a counterpart to Bismut's asymptotic formula in small time [3]. Contents 1. Introduction 1 2. The case of real hyperbolic spaces 4 2.1. Asymptotics of first-passage times for CIR-type diffusions 4 2.2. Three further estimates 10 2.3. End of the proof 12 3. The case of rank-one noncompact symmetric spaces 14 3.1. Some features of rank-one noncompact symmetric spaces 14 3.2. Proof of the theorem 15 3.3. A counterexample in rank two 23 4. The case of pinched Cartan-Hadamard manifolds 24 References 31
L�vy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator
Inventiones Mathematicae, 1996
We establish, by simple semigroup arguments, a L6vy-Gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the Gaussian measure. This produces in particular a new proof of the Gaussian isoperimetric inequality. This isoperimetric inequality strengthens the classical logarithmic Sobolev inequality in this context. A local version for the heat kernel measures is also proved, which may then be extended into an isoperimetric inequality for the Wiener measure on the paths of a Riemannian manifold with bounded Ricci curvature.
Geodesics of Random Riemannian Metrics: Supplementary Material
This is supplementary material for the main Geodesics article by the authors. In Appendix A, we present some general results on the construction of Gaussian random fields. In Appendix B, we restate our Shape Theorem from [LW10], specialized to the setting of this article. In Appendix C, we state some straightforward consequences on the geometry of geodesics for a random metric. In Appendix D, we provide a rapid introduction to Riemannian geometry for the unfamiliar reader. In Appendix E, we present some analytic estimates which we use in the article. In Appendix F, we present the construction of the conditional mean operator for Gaussian measures. In Appendix G, we describe Fermi normal coordinates, which we use in our construction of the bump metric.