Search results for "probability"
showing 10 items of 3417 documents
Optimal transport on the classical Wiener space with different norms
2011
In this paper we study two basic facts of optimal transportation on Wiener space W. Our first aim is to answer to the Monge Problem on the Wiener space endowed with the Sobolev type norm (k,gamma) to the power of p (cases p = 1 and p > 1 are considered apart). The second one is to prove 1-convexity (resp. C-convexity) along (constant speed) geodesics of relative entropy in (P2(W);W2), where W is endowed with the infinite norm (resp. with (k,gamma) norm), and W2 is the 2-distance of Wasserstein.
Maxima of a Random Model of the Riemann Zeta Function over Intervals of Varying Length
2021
We consider a model of the Riemann zeta function on the critical axis and study its maximum over intervals of length $(\log T)^{\theta}$, where $\theta$ is either fixed or tends to zero at a suitable rate. It is shown that the deterministic level of the maximum interpolates smoothly between the ones of log-correlated variables and of i.i.d. random variables, exhibiting a smooth transition 'from $\frac34$ to $\frac14$' in the second order. This provides a natural context where extreme value statistics of log-correlated variables with time-dependent variance and rate occur. A key ingredient of the proof is a precise upper tail tightness estimate for the maximum of the model on intervals of si…
Conical upper density theorems and porosity of measures
2008
Abstract We study how measures with finite lower density are distributed around ( n − m ) -planes in small balls in R n . We also discuss relations between conical upper density theorems and porosity. Our results may be applied to a large collection of Hausdorff and packing type measures.
Stochastic anticipative calculus on the path space over a compact Riemannian manifold
1998
Abstract In this paper, we shall first give another expression for Cruzeiro-Malliavin structure equation, by means of the Skorohod integral. The torsion tensor with respect to the Markovian connection used in [CF] is computed. This is the key step to establish a Stroock-like formula of commutation on the derivative of the Skorohod integral, which enables us to prove an Ito formula. As an application, we shall give a maximal inequality for Skorohod integrals following [AN2].
Geometry and analysis of Dirichlet forms
2012
Let $ \mathscr E $ be a regular, strongly local Dirichlet form on $L^2(X, m)$ and $d$ the associated intrinsic distance. Assume that the topology induced by $d$ coincides with the original topology on $ X$, and that $X$ is compact, satisfies a doubling property and supports a weak $(1, 2)$-Poincar\'e inequality. We first discuss the (non-)coincidence of the intrinsic length structure and the gradient structure. Under the further assumption that the Ricci curvature of $X$ is bounded from below in the sense of Lott-Sturm-Villani, the following are shown to be equivalent: (i) the heat flow of $\mathscr E$ gives the unique gradient flow of $\mathscr U_\infty$, (ii) $\mathscr E$ satisfies the Ne…
Heat semi-group and generalized flows on complete Riemannian manifolds
2011
Abstract We will use the heat semi-group to regularize functions and vector fields on Riemannian manifolds in order to develop Di Perna–Lions theory in this setting. Malliavinʼs point of view of the bundle of orthonormal frames on Brownian motions will play a fundamental role. As a byproduct we will construct diffusion processes associated to an elliptic operator with singular drift.
Sets versus trial sequences, Hausdorff versus von Mises: “Pure” mathematics prevails in the foundations of probability around 1920
2010
Abstract The paper discusses the tension which occurred between the notions of set (with measure) and (trial-) sequence (or—to a certain degree—between nondenumerable and denumerable sets) when used in the foundations of probability theory around 1920. The main mathematical point was the logical need for measures in order to describe general nondiscrete distributions, which had been tentatively introduced before (1919) based on von Mises’s notion of the “Kollektiv.” In the background there was a tension between the standpoints of pure mathematics and “real world probability” (in the words of J.L. Doob) at the time. The discussion and publication in English translation (in Appendix ) of two …
Numerical study of the transverse stability of the Peregrine solution
2020
We generalise a previously published approach based on a multi-domain spectral method on the whole real line in two ways: firstly, a fully explicit 4th order method for the time integration, based on a splitting scheme and an implicit Runge--Kutta method for the linear part, is presented. Secondly, the 1D code is combined with a Fourier spectral method in the transverse variable both for elliptic and hyperbolic NLS equations. As an example we study the transverse stability of the Peregrine solution, an exact solution to the one dimensional nonlinear Schr\"odinger (NLS) equation and thus a $y$-independent solution to the 2D NLS. It is shown that the Peregrine solution is unstable against all…
A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term
2017
This paper concerns the boundary behavior of solutions of certain fully nonlinear equations with a general drift term. We elaborate on the non-homogeneous generalized Harnack inequality proved by the second author in (Julin, ARMA -15), to prove a generalized Carleson estimate. We also prove boundary H\"older continuity and a boundary Harnack type inequality.
Sobolev estimates for optimal transport maps on Gaussian spaces
2012
We will study variations in Sobolev spaces of optimal transport maps with the standard Gaussian measure as the reference measure. Some dimension free inequalities will be obtained. As application, we construct solutions to Monge-Ampere equations in finite dimension, as well as on the Wiener space.