Chernoff's Theorem and Discrete Time Approximations of Brownian Motion on Manifolds

Let (S(t)) be a one-parameter family S = (S(t)) of positive integral operators on a locally compact space L. For a possibly non-uniform partition of [0,1] define a measure on the path space C([0,1],L) by using a) S(dt) for the transition between cosecutive partition times of distance dt, and b) a suitable continuous interpolation scheme (e.g. Brownian bridges or geodesics). If necessary normalize to get a probability measure. We prove a version of Chernoff's theorem of semigroup theory and tighness results which together yield convergence in law of such measures as the partition gets finer. In particular let L be a closed smooth submanifold of a Riemannian manifold M. We prove convergence of Brownian motion on M, conditioned to visit L at all partition times, to a process on L whose law has a Radon-Nikodym density with repect to Brownian motion on L which contains scalar, mean and sectional curvature terms. Various approximation schemes for Brownian motion are also given. These results substantially extend earlier work by the authors and by Andersson and Driver.Comment: 35 pages, revised version for publication, more detailed expositio

Analyticity and Riesz basis property of semigroups associated to damped vibrations

Second order equations of the form $z'' + A_0 z + D z'=0$ in an abstract Hilbert space are considered. Such equations are often used as a model for transverse motions of thin beams in the presence of damping. We derive various properties of the operator matrix $A$ associated with the second order problem above. We develop sufficient conditions for analyticity of the associated semigroup and for the existence of a Riesz basis consisting of eigenvectors and associated vectors of $A$ in the phase space

Extremal norms for positive linear inclusions

For finite-dimensional linear semigroups which leave a proper cone invariant it is shown that irreducibility with respect to the cone implies the existence of an extremal norm. In case the cone is simplicial a similar statement applies to absolute norms. The semigroups under consideration may be generated by discrete-time systems, continuous-time systems or continuous-time systems with jumps. The existence of extremal norms is used to extend results on the Lipschitz continuity of the joint spectral radius beyond the known case of semigroups that are irreducible in the representation theory interpretation of the word

Rates in the Central Limit Theorem and diffusion approximation via Stein's Method

We present a way to use Stein's method in order to bound the Wasserstein distance of order $2$ between two measures $\nu$ and $\mu$ supported on $\mathbb{R}^d$ such that $\mu$ is the reversible measure of a diffusion process. In order to apply our result, we only require to have access to a stochastic process $(X_t)_{t \geq 0}$ such that $X_t$ is drawn from $\nu$ for any $t > 0$. We then show that, whenever $\mu$ is the Gaussian measure $\gamma$, one can use a slightly different approach to bound the Wasserstein distances of order $p \geq 1$ between $\nu$ and $\gamma$ under an additional exchangeability assumption on the stochastic process $(X_t)_{t \geq 0}$. Using our results, we are able to obtain convergence rates for the multi-dimensional Central Limit Theorem in terms of Wasserstein distances of order $p \geq 2$. Our results can also provide bounds for steady-state diffusion approximation, allowing us to tackle two problems appearing in the field of data analysis by giving a quantitative convergence result for invariant measures of random walks on random geometric graphs and by providing quantitative guarantees for a Monte Carlo sampling algorithm

Combinatorial stability of non-deterministic systems

We introduce and study, from a combinatorial-topological viewpoint, some semigroups of continuous non-deterministic dynamical systems. Combinatorial stability, i.e. the persistence of the combinatorics of the attractors, is characterized and its genericity established. Some implications on topological (deterministic) dynamics are drawn

Estimating long term behavior of flows without trajectory integration: the infinitesimal generator approach

The long-term distributions of trajectories of a flow are described by invariant densities, i.e. fixed points of an associated transfer operator. In addition, global slowly mixing structures, such as almost-invariant sets, which partition phase space into regions that are almost dynamically disconnected, can also be identified by certain eigenfunctions of this operator. Indeed, these structures are often hard to obtain by brute-force trajectory-based analyses. In a wide variety of applications, transfer operators have proven to be very efficient tools for an analysis of the global behavior of a dynamical system. The computationally most expensive step in the construction of an approximate transfer operator is the numerical integration of many short term trajectories. In this paper, we propose to directly work with the infinitesimal generator instead of the operator, completely avoiding trajectory integration. We propose two different discretization schemes; a cell based discretization and a spectral collocation approach. Convergence can be shown in certain circumstances. We demonstrate numerically that our approach is much more efficient than the operator approach, sometimes by several orders of magnitude

Groups of given intermediate word growth

We show that there exists a finitely generated group of growth ~f for all functions f:\mathbb{R}\rightarrow\mathbb{R} satisfying f(2R) \leq f(R)^{2} \leq f(\eta R) for all R large enough and \eta\approx2.4675 the positive root of X^{3}-X^{2}-2X-4. This covers all functions that grow uniformly faster than \exp(R^{\log2/\log\eta}). We also give a family of self-similar branched groups of growth ~\exp(R^\alpha) for a dense set of \alpha\in(\log2/\log\eta,1).Comment: small typos corrected from v