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Marc Arnaudon

Publications and source records attributed to Marc Arnaudon.

46 records · Page 3Linked to original sources
arXiv2009

Horizontal diffusion in $C^1$ path space

We define horizontal diffusion in $C^1$ path space over a Riemannian manifold and prove its existence. If the metric on the manifold is developing under the forward Ricci flow, horizontal diffusion along Brownian motion turns out to be length preserving. As application, we prove contraction properties in the Monge-Kantorovich minimization problem for probability measures evolving along the heat flow. For constant rank diffusions, differentiating a family of coupled diffusions gives a derivative process with a covariant derivative of finite variation. This construction provides an alternative method to filtering out redundant noise.

math.PR↗
arXiv2009

Li-Yau Type Gradient Estimates and Harnack Inequalities by Stochastic Analysis

In this paper we use methods from Stochastic Analysis to establish Li-Yau type estimates for positive solutions of the heat equation. In particular, we want to emphasize that Stochastic Analysis provides natural tools to derive local estimates in the sense that the gradient bound at given point depends only on universal constants and the geometry of the Riemannian manifold locally about this point.

math.PR↗
arXiv2008

Existence of non-trivial harmonic functions on Cartan-Hadamard manifolds of unbounded curvature

The Liouville property of a complete Riemannian manifold (i.e., the question whether there exist non-trivial bounded harmonic functions) attracted a lot of attention. For Cartan-Hadamard manifolds the role of lower curvature bounds is still an open problem. We discuss examples of Cartan-Hadamard manifolds of unbounded curvature where the limiting angle of Brownian motion degenerates to a single point on the sphere at infinity, but where nevertheless the space of bounded harmonic functions is as rich as in the non-degenerate case. To see the full boundary the point at infinity has to be blown up in a non-trivial way. Such examples indicate that the situation concerning the famous conjecture of Greene and Wu about existence of non-trivial bounded harmonic functions on Cartan-Hadamard manifolds is much more complicated than one might have expected.

math.PR↗
arXiv2008

Convex ordering for random vectors using predictable representation

We prove convex ordering results for random vectors admitting a predictable representation in terms of a Brownian motion and a non-necessarily independent jump component. Our method uses forward-backward stochastic calculus and extends previous results in the one-dimensional case. We also study a geometric interpretation of convex ordering for discrete measures in connection with the conditions set on the jump heights and intensities of the considered processes.

math.PR↗
arXiv2007

Information Criteria and Arithmetic Codings : An Illustration on Raw Images

In this paper we give a short theoretical description of the general predictive adaptive arithmetic coding technique. The links between this technique and the works of J. Rissanen in the 80's, in particular the BIC information criterion used in parametrical model selection problems, are established. We also design lossless and lossy coding techniques of images. The lossless technique uses a mix between fixed-length coding and arithmetic coding and provides better compression results than those separate methods. That technique is also seen to have an interesting application in the domain of statistics since it gives a data-driven procedure for the non-parametrical histogram selection problem. The lossy technique uses only predictive adaptive arithmetic codes and shows how a good choice of the order of prediction might lead to better results in terms of compression. We illustrate those coding techniques on a raw grayscale image.

cs.IT↗
arXiv2007

Codage arithmetique pour la description d'une distribution

Using predictive adaptive arithmetic coding and the Minimum Description Length principle, we derive an efficient tool for model selection problems : the RIC information criterion. We then present an extension of these coding techniques to non-parametrical estimation of a distribution and illustrate it on the gray scales histogram of an image. Key-words : Information criteria, MDL, model selection, non-parametrical estimation, histograms.

stat.ME↗
arXiv2007

An improved method for model selection based on Information Criteria

Information criteria are an appropriate and widely used tool for solving model selection problems. However, different ways to use them exist, each leading to a more or less precise approximation of the sought model. In this paper, we mainly present two methods of utilisation of information criteria : the classical one which is generally used and an alternative one, more precise but requiring a little more calculations. Those methods are compared on 1-D and 2-D autoregressive models ; we use a synthetized process for the 1-D case and texture images for the 2-D case. We also work with the original phi_beta criterion which includes all others usual criteria such as AIC, BIC, and phi.

math.ST↗
arXiv2005

Barycenters of measures transported by stochastic flows

We investigate the evolution of barycenters of masses transported by stochastic flows. The state spaces under consideration are smooth affine manifolds with certain convexity structure. Under suitable conditions on the flow and on the initial measure, the barycenter {Z_t} is shown to be a semimartingale and is described by a stochastic differential equation. For the hyperbolic space the barycenter of two independent Brownian particles is a martingale and its conditional law converges to that of a Brownian motion on the limiting geodesic. On the other hand for a large family of discrete measures on suitable Cartan-Hadamard manifolds, the barycenter of the measure carried by an unstable Brownian flow converges to the Busemann barycenter of the limiting measure.

math.PR↗
arXiv2004

Concentration of the Brownian bridge on Cartan-Hadamard manifolds with pinched negative sectional curvature

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, and one of us. 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.

math.PR↗