Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes

The present work extends our short communication Phys. Rev. Lett. 95, 111102 (2005). For smooth marginally outer trapped surfaces (MOTS) in a smooth spacetime we define stability with respect to variations along arbitrary vectors v normal to the MOTS. After giving some introductory material about linear non self-adjoint elliptic operators, we introduce the stability operator L_v and we characterize stable MOTS in terms of sign conditions on the principal eigenvalue of L_v. The main result shows that given a strictly stable MOTS S contained in one leaf of a given reference foliation in a spacetime, there is an open marginally outer trapped tube (MOTT), adapted to the reference foliation, which contains S. We give conditions under which the MOTT can be completed. Finally, we show that under standard energy conditions on the spacetime, the MOTT must be either locally achronal, spacelike or null.Comment: 33 pages, no figures, typos corrected, minor changes in presentatio

Limits of Random Differential Equations on Manifolds

Consider a family of random ordinary differential equations on a manifold driven by vector fields of the form $\sum_kY_k\alpha_k(z_t^\epsilon(\omega))$ where $Y_k$ are vector fields, $\epsilon$ is a positive number, $z_t^\epsilon$ is a ${1\over \epsilon} {\mathcal L}_0$ diffusion process taking values in possibly a different manifold, $\alpha_k$ are annihilators of $ker ({\mathcal L}_0^*)$. Under H\"ormander type conditions on ${\mathcal L}_0$ we prove that, as $\epsilon$ approaches zero, the stochastic processes $y_{t\over \epsilon}^\epsilon$ converge weakly and in the Wasserstein topologies. We describe this limit and give an upper bound for the rate of the convergence.Comment: 46 pages, To appear in Probability Theory and Related Fields In this version, we add a note in proof for the published versio

Isoperimetric, Sobolev, and eigenvalue inequalities via the Alexandroff-Bakelman-Pucci method: A survey

This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric inequality in convex cones. For this last inequality, the new proof was recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with weights came up by studying an open question raised by Haim Brezis.Peer ReviewedPostprint (published version

On small time asymptotics for rough differential equations driven by fractional Brownian motions

We survey existing results concerning the study in small times of the density of the solution of a rough differential equation driven by fractional Brownian motions. We also slightly improve existing results and discuss some possible applications to mathematical finance.Comment: This is a survey paper, submitted to proceedings in the memory of Peter Laurenc

Asymptotic behavior of divergences and Cameron-Martin theorem on loop spaces

We first prove the L^p-convergence (p\geq 1) and a Fernique-type exponential integrability of divergence functionals for all Cameron-Martin vector fields with respect to the pinned Wiener measure on loop spaces over a compact Riemannian manifold. We then prove that the Driver flow is a smooth transform on path spaces in the sense of the Malliavin calculus and has an \infty-quasi-continuous modification which can be quasi-surely well defined on path spaces. This leads us to construct the Driver flow on loop spaces through the corresponding flow on path spaces. Combining these two results with the Cruzeiro lemma [J. Funct. Anal. 54 (1983) 206-227] we give an alternative proof of the quasi-invariance of the pinned Wiener measure under Driver's flow on loop spaces which was established earlier by Driver [Trans. Amer. Math. Soc. 342 (1994) 375-394] and Enchev and Stroock [Adv. Math. 119 (1996) 127-154] by Doob's h-processes approach together with the short time estimates of the gradient and the Hessian of the logarithmic heat kernel on compact Riemannian manifolds. We also establish the L^p-convergence (p\geq 1) and a Fernique-type exponential integrability theorem for the stochastic anti-development of pinned Brownian motions on compact Riemannian manifold with an explicit exponential exponent. Our results generalize and sharpen some earlier results due to Gross [J. Funct. Anal. 102 (1991) 268-313] and Hsu [Math. Ann. 309 (1997) 331-339]. Our method does not need any heat kernel estimate and is based on quasi-sure analysis and Sobolev estimates on path spaces.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000004

Maximal regularity and Hardy spaces

In this work, we consider the Cauchy problem for $u' - Au = f$ with $A$ the Laplacian operator on some Riemannian manifolds or a sublapacian on some Lie groups or some second order elliptic operators on a domain. We show the boundedness of the operator of maximal regularity $f\mapsto Au$ and its adjoint on appropriate Hardy spaces which we define and study for this purpose. As a consequence we reobtain the maximal $L^q$ regularity on $L^p$ spaces for $p,q$ between 1 and $\infty$.Comment: 27 page