Brownian Motion and Negative Curvature

Mathematische Zeitschrift, 1989

Transactions of the American Mathematical Society, 1993

Let M M be the universal cover of a compact nonflat surface N N of nonpositive curvature. We show that on the average the Brownian motion on M M behaves similarly to the Brownian motion on negatively curved manifolds. We use this to prove that harmonic measures on the sphere at infinity have positive Hausdorff dimension and if the geodesic flow on N N is ergodic then the harmonic and geodesic measure classes at infinity are singular unless the curvature is constant.

Pacific Journal of Mathematics, 1981

Potential Analysis, 2017

We introduce and study Brownian bridges to submanifolds. Our method involves proving a general formula for the integral over a submanifold of the minimal heat kernel on a complete Riemannian manifold. We use the formula to derive lower bounds, an asymptotic relation and derivative estimates. We also see a connection to hypersurface local time. This work is motivated by the desire to extend the analysis of path and loop spaces to measures on paths which terminate on a submanifold.

Advances in Mathematics, 1984

Revista Matemática Iberoamericana, 2002

The infinite Brownian loop {B 0 t , t ≥ 0} on a Riemannian manifold M is the limit in distribution of the Brownian bridge of length T around a fixed origin 0, when T → +∞. It has no spectral gap. When M has nonnegative Ricci curvature, B 0 is the Brownian motion itself. When M = G/K is a noncompact symmetric space, B 0 is the relativized Φ 0-process of the Brownian motion, where Φ 0 denotes the basic spherical function of Harish-Chandra, i.e. the K-invariant ground state of the Laplacian. In this case, we consider the polar decomposition B 0 t = (K t , X t), where K t ∈ K/M and X t ∈ā + , the positive Weyl chamber. Then, as t → +∞, K t converges and d(0, X t)/t → 0 almost surely. Moreover the processes {X tT / √ T , t ≥ 0} converge in distribution, as T → +∞, to the intrinsic Brownian motion of the Weyl chamber. This implies in particular that d(0, X tT)/ √ T converges to a Bessel process of dimension D = rank M + 2j, where j denotes the number of positive indivisible roots. An ingredient of the proof is a new estimate on Φ 0 .

Stochastic Processes and their Applications, 2016

In this paper we introduce three Markovian couplings of Brownian motions on smooth Riemannian manifolds without boundary which sit at the crossroad of two concepts. The first concept is the one of shy coupling put forward in [3] and the second concept is the lower bound on the Ricci curvature and the connection with couplings made in [30]. The first construction is the shy coupling, the second one is a fixed-distance coupling and the third is a coupling in which the distance between the processes is a deterministic exponential function of time. The result proved here is that an arbitrary Riemannian manifold satisfying some technical conditions supports shy couplings. If in addition, the Ricci curvature is non-negative, there exist fixed-distance couplings. Furthermore, if the Ricci curvature is bounded below by a positive constant, then there exists a coupling of Brownian motions for which the distance between the processes is a decreasing exponential function of time. The constructions use the intrinsic geometry, and relies on an extension of the notion of frames which plays an important role for even dimensional manifolds. In fact, we provide a wider class of couplings in which the distance function is deterministic in Theorem 5 and Corollary 9. As an application of the fixed-distance coupling we derive a maximum principle for the gradient of harmonic functions on manifolds with non-negative Ricci curvature. As far as we are aware of, these constructions are new, though the existence of shy couplings on manifolds is suggested by Kendall in [17].

We study the geometrical influence on the Brownian motion over curved manifolds. We focus on the following intriguing question: what observables are appropriated to measure Brownian motion in curved manifolds? In particular, for those d-dimensional manifolds embedded in $\mathbb{R}^{d+1}$ we define three quantities for the displacement's notion, namely, the geodesic displacement, $s$, the Euclidean displacement, $\deltaR$, and the projected Euclidean displacement $\deltaR_{\perp}$. In addition, we exploit the Weingarten-Gauss equations in order to calculate the mean-square Euclidean displacement's in the short-time regime. Besides, it is possible to prove exact formulas for these expectation values, at all times, in spheres and minimal hypersurfaces. In the latter case, Brownian motion corresponds to the typical diffusion in flat geometries, albeit minimal hypersurfaces are not intrinsically flat. Finally, the two-dimensional case is emphasized since its relation to the late...

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 2001

We consider the Brownian bridge of length T on a symmetric space of the noncompact type. We prove that this process, properly rescaled, converges when T → +∞ to a process whose generalized radial part is the bridge of the Euclidean Brownian motion in the Weyl chamber killed at the boundary.  2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Le pont brownien sur les espaces riemanniens symétriques Résumé. On considère le pont brownien de longueur T sur un espace symétrique de type non compact. On montre que ce processus, convenablement renormalisé, converge lorsque T → +∞ vers un processus dont la partie radiale généralisée est le pont du mouvement brownien dans la chambre de Weyl tué à la frontière.  2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Version française abrégée Considérons un espace riemannien symétrique de type non compact M. Par définition M = G/K où G est un groupe semi-simple connexe de centre fini et K est un sous-groupe compact maximal. Soit g = k + p la décomposition de Cartan de l'algèbre de Lie g de G. On choisit un sous espace abélien maximal a de p et on le munit de la structure euclidienne donnée par la forme de Killing. Soit a + une chambre de Weyl de a et Σ + 0 l'ensemble des racines indivisibles et positives associées. Rappelons la décomposition polaire généralisée de M. On choisit o = K comme origine dans M. Soit A = exp a et soit M le centralisateur de A dans K. Pour x ∈ M, soitk(x) ∈ K/M et C(x) ∈ a + tels que k(x) e C(x) • o = x, où k(x) ∈ K est un représentant dek(x). Nous utilisons aussi la décomposition d'Iwasawa G = KN A. Chaque g ∈ G s'écrit g = K(g)N (g) e H(g) , où K(g) ∈ K, N (g) ∈ N et H(g) ∈ a. Le mouvement brownien B sur M est le processus de Markov de générateur ∆/2. Son semi-groupe admet des densités p t (x, y) strictement positives et symétriques par rapport à la mesure riemannienne m. Le pont de a ∈ M à b ∈ M de longueur T est le processus de Markov B {a,b,T } t , 0 t T non homogène, Note présentée par Marc YOR. S0764-4442(01)02145-0/FLA  2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés

2021

We prove a number of results relating exit times of planar Brownian with the geometric properties of the domains in question. Included are proofs of the conformal invariance of moduli of rectangles and annuli using Brownian motion; similarly probabilistic proofs of some recent results of Karafyllia on harmonic measure on starlike domains; examples of domains and their complements which are simultaneously large when measured by the moments of exit time of Brownian motion, and examples of domains and their complements which are simultaneously small; and proofs of several identities involving the Cauchy distribution using the optional stopping theorem.