The purpose of this work is to construct a {\it Brownian motion} with values in simplicial complexes with piecewise differential structure. In order to state and prove the existence of such Brownian motion, we define a family of continuous Markov processes with values in an admissible complex; we call every process of this family, {\it isotropic transport process}. We show that the family of the isotropic processes contains a subsequence, which converges weakly to a measure; we name it the {\it Wiener measure}. Then, using the finite dimensional distributions of the obtained Wiener measure, we construct a new admissible complex valued continuous Markov process: the Brownian motion. We finished with a geometric analysis of this Brownian motion, to determine the recurrent or transient behavior of such process.
We show that the pointed measured Gromov convergence of the underlying spaces implies (or under some condition, is equivalent to) the weak convergence of Brownian motions under Riemannian Curvature-Dimension (RCD) conditions.
We prove large deviations for $g(t)$-Brownian motion in a complete, evolving Riemannian manifold $M$ with respect to a collection $\{g(t)\}_{t\in [0,1]}$ of Riemannian metrics, smoothly depending on $t$. We show how the large deviations are obtained from the large deviations of the (time-dependent) horizontal lift of $g(t)$-Brownian motion to the frame bundle $FM$ over $M$. The latter is proved by embedding the frame bundle into some Euclidean space and applying Freidlin-Wentzell theory for diffusions with time-dependent coefficients, where the coefficients are jointly Lipschitz in space and t…
We define kinetic Brownian motion on the diffeomorphism group of a closed Riemannian manifold, and prove that it provides an interpolation between the hydrodynamic flow of a fluid and a Brownian-like flow.
We give an effective upper escape rate function for Brownian motion on a complete Riemannian manifold in terms of the volume growth of the manifold. An important step in the work is estimating the small tail probability of the crossing time between two concentric geodesic spheres by reflecting Brownian motions on the larger geodesic ball.
We study the radial part of sub-Riemannian Brownian motion in the context of totally geodesic foliations. Itô's formula is proved for the radial processes associated to Riemannian distances approximating the Riemannian one. We deduce very general stochastic completeness criteria for the sub-Riemannian Brownian motion. In the context of Sasakian foliations and H-type groups, one can push the analysis further, and taking advantage of the recently proved sub-Laplacian comparison theorems one can compare the radial processes for the sub-Riemannian distance to one-dimensional model diffusions. As a…
We consider in this work a one parameter family of hypoelliptic diffusion processes on the unit tangent bundle $T^1 \mathcal M$ of a Riemannian manifold $(\mathcal M,g)$, collectively called kinetic Brownian motions, that are random perturbations of the geodesic flow, with a parameter $σ$ quantifying the size of the noise. Projection on $\mathcal M$ of these processes provides random $C^1$ paths in $\mathcal M$. We show, both qualitively and quantitatively, that the laws of these $\mathcal M$-valued paths provide an interpolation between geodesic and Brownian motions. This qualitative descript…
We use geometric microlocal methods to compute an asymptotic expansion of mean first arrival time for Brownian particles on Riemannian manifolds. This approach provides a robust way to treat this problem, which has thus far been limited to very special geometries. This paper can be seen as the Riemannian 3-manifold version of the planar result of \cite{ammari} and thus enable us to see the full effect of the local extrinsic boundary geometry on the mean arrival time of the Brownian particles. Our approach also connects this question to some of the recent progress on boundary rigidity and integ…
We generalize Brownian motion on a Riemannian manifold to the case of a family of metrics which depends on time. Such questions are natural for equations like the heat equation with respect to time dependent Laplacians (inhomogeneous diffusions). In this paper we are in particular interested in the Ricci flow which provides an intrinsic family of time dependent metrics. We give a notion of parallel transport along this Brownian motion, and establish a generalization of the Dohrn-Guerra or damped parallel transport, Bismut integration by part formulas, and gradient estimate formulas. One of our…
This paper considers a classical question of approximation of Brownian motion by a random walk in the setting of a sub-Riemannian manifold $M$. To construct such a random walk we first address several issues related to the degeneracy of such a manifold. In particular, we define a family of sub-Laplacian operators naturally connected to the geometry of the underlining manifold. In the case when $M$ is a Riemannian (non-degenerate) manifold, we recover the Laplace-Beltrami operator. We then construct the corresponding random walk, and under standard assumptions on the sub-Laplacian and $M$ we sh…
We prove the convergence of the spectrum of the generator of the kinetic Brownian motion to the spectrum of the base Laplacian for closed Riemannian manifolds. This generalizes recent work of Kolb--Weich--Wolf [arXiv:2011.06434] on constant curvature surfaces and of Ren--Tao [arXiv:2208.13111] on locally symmetric spaces. As an application, we prove a conjecture of Baudoin--Tardif [arXiv:1604.06813] on the optimal convergence rate to the equilibrium.
Analogous to the characterisation of Brownian motion on a Riemannian manifold as the development of Brownian motion on a Euclidean space, we construct sub-Riemannian diffusions on equinilpotentisable sub-Riemannian manifolds by developing a canonical stochastic process arising as the lift of Brownian motion to an associated model space. The notion of stochastic development we introduce for equinilpotentisable sub-Riemannian manifolds uses Cartan connections, which take the place of the Levi-Civita connection in Riemannian geometry. We first derive a general expression for the generator of the …
We study the problem of optimal dividend payout from a surplus process governed by Brownian motion with drift under the additional constraint of ratcheting, i.e. the dividend rate can never decrease. We solve the resulting two-dimensional optimal control problem, identifying the value function to be the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. For finitely many admissible dividend rates we prove that threshold strategies are optimal, and for any finite continuum of admissible dividend rates we establish the $\varepsilon$-optimality of curve strategies. T…
We use reflecting Brownian motion (RBM) to prove the well known Gauss-Bonnet-Chern theorem for a compact Riemannian manifold with boundary. The boundary integrand is obtained by carefully analyzing the asymptotic behavior of the boundary local time of RBM for small times.
The infinite Brownian loop on a Riemannian manifold is the limit in distribution of the Brownian bridge of length $T$ around a fixed origin when $T \rightarrow +\infty$. The aim of this note is to study its long-time asymptotics on Riemannian symmetric spaces $G/K$ of noncompact type and of general rank. This amounts to the behavior of solutions to the heat equation subject to the Doob transform induced by the ground spherical function. Unlike the standard Brownian motion, we observe in this case phenomena which are similar to the Euclidean setting, namely $L^1$ asymptotic convergence without …
Extensions of the fractional Brownian fields are constructed over a complete Riemannian manifold. This construction is carried out for the full range of the Hurst parameter $α\in(0,1)$. In particular, we establish existence, distributional scaling (self-similiarity), stationarity of the increments, and almost sure Hölder continuity of sample paths. Stationary counterparts to these fields are also constructed.
We define the Ricci curvature of Markov chains on metric spaces as a local contraction coefficient of the random walk acting on the space of probability measures equipped with a Wasserstein transportation distance. For Brownian motion on a Riemannian manifold this gives back the value of Ricci curvature of a tangent vector. Examples of positively curved spaces for this definition include the discrete cube and discrete versions of the Ornstein--Uhlenbeck process. Moreover this generalization is consistent with the Bakry--Émery Ricci curvature for Brownian motion with a drift on a Riemannian man…
We study a class of mean-field stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H\in(1/2,1)$ and a related stochastic control problem. We derive a Pontryagin type maximum principle and the associated adjoint mean-field backward stochastic differential equation driven by a classical Brownian motion, and we prove that under certain assumptions, which generalise the classical ones, the necessary condition for the optimality of an admissible control is also sufficient.
By further developing the generalized $Γ$-calculus for hypoelliptic operators, we prove hypocoercive estimates for a large class of Kolmogorov type operators which are defined on non necessarily totally geodesic Riemannian foliations. We study then in detail the example of the velocity spherical Brownian motion, whose generator is a step-3 generating hypoelliptic Hörmander's type operator. To prove hypocoercivity in that case, the key point is to show the existence of a convenient Riemannian foliation associated to the diffusion. We will then deduce, under suitable geometric conditions, the co…
We prove the strong convergence of the spectrum of the kinetic Brownian motion to the spectrum of base Laplacian for a large class of compact locally Riemannian homogeneous spaces, in particular all compact locally symmetric spaces. This generalizes recent work of Kolb--Weich--Wolf [arXiv:2011.06434] on constant curvature surfaces.
The purpose of this note is to give details for an argument of Sullivan to construct eigenfunctions of the Laplacian on a Riemannian manifold using exit times of Brownian motion \cite{sullivanpos}. Let $X$ be a complete, simply connected Riemannian manifold of pinched negative sectional curvature. Let $λ_1 = λ_1(X) < 0$ be the supremum of the spectrum of the Laplacian on $L^2(X)$, and let $D \subset X$ be a bounded domain in $X$ with smooth boundary. Let $(B_t)_{t \geq 0}$ be Brownian motion on $X$ and let $τ= τ_D$ be the first exit time of Brownian motion from $D$. For each $λ\in \mathbb{C}$ …
Brownian motions on star graphs in the sense of Itô-McKean, that is, Walsh processes admitting a generalized boundary behavior including stickiness and jumps and having an angular distribution with finite support, are examined. Their generators are identified as Laplace operators on the graph subject to non-local Feller-Wentzell boundary conditions. A pathwise description is achieved for every admissible boundary condition: For finite jump measures, a construction of Kostrykin, Potthoff and Schrader in the continuous setting is expanded via a technique of successive killings and revivals; for …
Let $W(t), t\ge 0$ be standard Brownian motion. We study the size of the time intervals $I$ which are admissible for the long range of slow increase, namely given a real $z>0$, $$ \sup_{t\in I}{|W(t)|\over \sqrt t} \le z, $$ and we estimate their number of occurences. We obtain optimal results in terms of class test functions and, by means of the quantitative Borel-Cantelli lemma, a fine frequency result concerning their occurences. Using Sakhanenko's invariance principe to transfer the results to the Kubilius model, we derive applications to the prime number divisor function. We obtain refine…
We propose new copulae to model the dependence between two Brownian motions and to control the distribution of their difference. Our approach is based on the copula between the Brownian motion and its reflection. We show that the class of admissible copulae for the Brownian motions are not limited to the class of Gaussian copulae and that it also contains asymmetric copulae. These copulae allow for the survival function of the difference between two Brownian motions to have higher value in the right tail than in the Gaussian copula case. Considering two Brownian motions $B_t^1$ and $B_t^2$, th…
Let $(Z,κ)$ be a Walsh Brownian motion with spinning measure $κ$. Suppose $μ$ is a probability measure on $\mathbb{R}^n$. We characterize all the $κ$ such that $μ$ is a stopping distribution of $(Z,κ)$. If we further restrict the solution to be integrable, we show that there would be only one choice of $κ$. We also generalize Vallois' embedding, and prove that it minimizes the expectation $\mathbb{E}[Ψ(L^Z_τ)]$ among all the admissible solutions $τ$, where $Ψ$ is a strictly convex function and $(L_t^Z)_{t \geq 0}$ is the local time of the Walsh Brownian motion at the origin.
Let $\{g_t\}_{t\in [0,T)}$ be a family of complete time-depending Riemannian matrics on a manifold which evolves under backwards Ricci flow. The Itô formula is established for the $\mathcal{L}$-distance of the $g_t$-Brownian motion to a fixed reference point ($\mathcal{L}$-base). Furthermore, as an application, we construct a coupling by parallel displacement which yields a new proof of some results of Topping.
We show that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric.
Given a submersion $ϕ: M \to N$, where $M$ is Riemannian, we construct a stochastic process $X$ on $M$ such that the image $Y:=ϕ(X)$ is a (reversed, scaled) mean curvature flow of the fibers of the submersion. The model example is the mapping $π: GL(n) \to GL(n)/O(n)$, whose image is equivalent to the space of $n$-by-$n$ positive definite matrices, $\pdef$, and the said MCF has deterministic image. We are able to compute explicitly the mean curvature (and hence the drift term) of the fibers w.r.t. this map, (i) under diagonalization and (ii) in matrix entries, writing mean curvature as the gra…
We prove the central limit theorem of random variables induced by distances to Brownian paths and Green functions on the universal cover of Riemannian manifolds of finite volume with pinched negative curvature. We further provide some ergodic properties of Brownian motions and an application of the central limit theorem to the dynamics of geodesic flows in pinched negative curvature.
We show that on a manifold whose Riemannian metric evolves under backwards Ricci flow two Brownian motions can be coupled in such a way that the expectation of their normalized L-distance is non-increasing. As an immediate corollary we obtain a new proof of a recent result of Topping (J. reine angew. Math. 636 (2009), 93-122), namely that the normalized L-transportation cost between two solutions of the heat equation is non-increasing as well.
Let $Q$ be a Riemannian $G$-manifold. This paper is concerned with the symmetry reduction of Brownian motion in $Q$ and ramifications thereof in a Hamiltonian context. Specializing to the case of polar actions we discuss various versions of the stochastic Hamilton-Jacobi equation associated to the symmetry reduction of Brownian motion and observe some similarities to the Schrödinger equation of the quantum free particle reduction as described by Feher and Pusztai. As an application we use this reduction scheme to derive examples of quantum Calogero-Moser systems from a stochastic setting.
We prove tightness of a family of path measures $ν_{\varepsilon}$ on tubes $L(\varepsilon)$ of small diameters around a closed and connected submanifold $L$ of another Riemannian manifold $M$. Together with a convergence result for Dirichlet semigroups on tubular neighbourhoods, that implies weak convergence of the measures as the tube radius $\varepsilon$ tends to zero to a measure supported by the path space of the submanifold. As a consequence, we obtain weak convergence of the measures obtained by conditioning Brownian motion to stay within the tubes $L(\varepsilon)$ up to a finite time $T…
We consider the symmetric exclusion process on suitable random grids that approximate a compact Riemannian manifold. We prove that a class of random walks on these random grids converge to Brownian motion on the manifold. We then consider the empirical density field of the symmetric exclusion process and prove that it converges to the solution of the heat equation on the manifold.
In this article, we consider flat and curved Riemannian symmetric spaces in the complex case and we study their basic integral kernels, in potential and spherical analysis: heat, Newton, Poisson kernels and spherical functions, i.e. the kernel of the spherical Fourier transform. We introduce and exploit a simple new method of construction of these $W$-invariant kernels by alternating sum formulas. We then use the alternating sum representation of these kernels to obtain their asymptotic behavior. We apply our results to the Dyson Brownian Motion on $R^d$.
The purpose of this paper is to construct a Brownian motion $X := (X_t)_{t\geq 0}$ taking values in a Riemannian manifold $M$, together with a compact valued process $D:= (D_t)_{t\geq 0}$ such that, at least for small enough ${\mathscr F}^D$-stopping time $τ> 0$ and conditioned by ${\mathscr F}_τ^D$, the law of $X_τ$ is the normalized Lebesgue measure on $D_τ$. This intertwining result is a generalization of Pitman theorem. We first construct regular intertwined processes related to Stokes' theorem. Then using several limiting procedures we construct synchronous intertwined, free intertwined, …
The kinetic Brownian motion on the cosphere bundle of a Riemannian manifold $\mathbb{M}$ is a stochastic process that models the geodesic equation perturbed by a random white force of size $\varepsilon$. When $\mathbb{M}$ is compact with negative curvature we show that the $L^2$-spectrum of the infinitesimal generator of this process converges to the Pollicott--Ruelle resonances of $\mathbb{M}$ as $\varepsilon \rightarrow 0$.
We study co-adapted couplings of (canonical hypoelliptic) diffu-sions on the (subRiemannian) Heisenberg group, that we call (Heisenberg) Brow-nian motions and are the joint laws of a planar Brownian motion with its L{é}vy area. We show that contrary to the situation observed on Riemannian manifolds of non-negative Ricci curvature, for any co-adapted coupling, two Heisenberg Brownian motions starting at two given points can not stay at bounded distance for all time t $\ge$ 0. Actually, we prove the stronger result that they can not stay bounded in L p for p $\ge$ 2. We also study the coupling b…
In this paper we study a random walk on an affine building of type $\tilde{A}_r$, whose radial part, when suitably normalized, converges to the Brownian motion of the Weyl chamber. This gives a new discrete approximation of this process, alternative to the one of Biane \cite{Bia2}. This extends also the link at the probabilistic level between Riemannian symmetric spaces of the noncompact type and their discrete counterpart, which had been previously discovered by Bougerol and Jeulin in rank one \cite{BJ}. The main ingredients of the proof are a combinatorial formula on the building and the est…
The present paper is devoted to the study of sample paths of G-Brownian motion and stochastic differential equations (SDEs) driven by G-Brownian motion from the view of rough path theory. As the starting point, we show that quasi-surely, sample paths of G-Brownian motion can be enhanced to the second level in a canonical way so that they become geometric rough paths of roughness 2 < p < 3. This result enables us to introduce the notion of rough differential equations (RDEs) driven by G-Brownian motion in the pathwise sense under the general framework of rough paths. Next we establish the funda…
We investigate yet another approach to understand the limit behaviour of Brownian motion conditioned to stay within a tubular neighbourhood around a closed and connected submanifold of a Riemannian manifold. In this context, we identify a second order generator subject to Dirichlet conditions on the boundary of the tube and study its associated semigroups. After a suitable rescaling and renormalization procedure, we obtain convergence of these semigroups, both in $L^2$ and in Sobolev spaces of arbitrarily large index, to a limit semigroup, as the tube diameter tends to zero. As a byproduct, we…
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 \cite{Burdzy-Benjamini} and the second concept is the lower bound on the Ricci curvature and the connection with couplings made in \cite{ReSt}. 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…
We present a study of the distance between a Brownian motion and a submanifold of a complete Riemannian manifold. We include a variety of results, including an inequality for the Laplacian of the distance function derived from a Jacobian comparison theorem, a characterization of local time on a hypersurface which includes a formula for the mean local time, an exit time estimate for tubular neighbourhoods and a concentration inequality. We derive the concentration inequality using moment estimates to obtain an exponential bound, which holds under fairly general assumptions and which is sufficie…
