Geometry of non-holonomic diffusion

Simon Hochgerner; Tudor S. Ratiu

Displaying similar documents to “Geometry of non-holonomic diffusion”

A well-posedness result for a mass conserved Allen-Cahn equation with nonlinear diffusion

Kettani, Perla El, Hilhorst, Danielle, Lee, Kai

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In this paper, we prove the existence and uniqueness of the solution of the initial boundary value problem for a stochastic mass conserved Allen-Cahn equation with nonlinear diffusion together with a homogeneous Neumann boundary condition in an open bounded domain of $ℝ^{n}$ with a smooth boundary. We suppose that the additive noise is induced by a Q-Brownian motion.

Transience, recurrence and speed of diffusions with a non-markovian two-phase “use it or lose it” drift

Ross G. Pinsky (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We investigate the transience/recurrence of a non-Markovian, one-dimensional diffusion process which consists of a Brownian motion with a non-anticipating drift that has two phases – a transient to $+ \infty$ mode which is activated when the diffusion is sufficiently near its running maximum, and a recurrent mode which is activated otherwise. We also consider the speed of a diffusion with a two-phase drift, where the drift is equal to a certain non-negative constant when the diffusion is sufficiently...

Superdiffusivity for brownian motion in a poissonian potential with long range correlation II: Upper bound on the volume exponent

Hubert Lacoin (2012)

Annales de l'I.H.P. Probabilités et statistiques

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This paper continues a study on trajectories of Brownian Motion in a field of soft trap whose radius distribution is unbounded. We show here that for both point-to-point and point-to-plane model the volume exponent (the exponent associated to transversal fluctuation of the trajectories) $ξ$ is strictly less than $1$ and give an explicit upper bound that depends on the parameters of the problem. In some specific cases, this upper bound matches the lower bound proved in the first part of this...

The brownian cactus I. Scaling limits of discrete cactuses

Nicolas Curien, Jean-François Le Gall, Grégory Miermont (2013)

Annales de l'I.H.P. Probabilités et statistiques

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The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space $E$ , one can associate an $ℝ$ -tree called the continuous cactus of $E$ . We prove under general assumptions that the cactus of random planar maps distributed according to Boltzmann weights and conditioned to have a fixed large number of vertices converges in distribution to a limiting space called the Brownian cactus, in the Gromov–Hausdorff sense. Moreover, the Brownian...

A note on one-dimensional stochastic equations

Hans-Jürgen Engelbert (2001)

Czechoslovak Mathematical Journal

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We consider the stochastic equation $X_{t} = x_{0} + \int_{0}^{t} b (u, X_{u}) d B_{u}, t \geq 0,$ where $B$ is a one-dimensional Brownian motion, $x_{0} \in ℝ$ is the initial value, and $b [0, \infty) \times ℝ \to ℝ$ is a time-dependent diffusion coefficient. While the existence of solutions is well-studied for only measurable diffusion coefficients $b$ , beyond the homogeneous case there is no general result on the uniqueness in law of the solution. The purpose of the present note is to give conditions on $b$ ensuring the existence as well as the uniqueness in law of the solution. ...

Optimal stopping with advanced information flow: selected examples

Yaozhong Hu, Bernt Øksendal (2008)

Banach Center Publications

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We study optimal stopping problems for some functionals of Brownian motion in the case when the decision whether or not to stop before (or at) time t is allowed to be based on the δ-advanced information $ℱ_{t + δ}$ , where $ℱ_{s}$ is the σ-algebra generated by Brownian motion up to time s, s ≥ -δ, δ > 0 being a fixed constant. Our approach involves the forward integral and the Malliavin calculus for Brownian motion.

The Dyson Brownian Minor Process

Mark Adler, Eric Nordenstam, Pierre Van Moerbeke (2014)

Annales de l’institut Fourier

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Consider an $n \times n$ Hermitean matrix valued stochastic process ${H_{t}}_{t \geq 0}$ where the elements evolve according to Ornstein-Uhlenbeck processes. It is well known that the eigenvalues perform a so called Dyson Brownian motion, that is they behave as Ornstein-Uhlenbeck processes conditioned never to intersect. In this paper we study not only the eigenvalues of the full matrix, but also the eigenvalues of all the principal minors. That is, the eigenvalues of the $k \times k$ minors in the upper left corner...

Existence and asymptotic behaviour of some time-inhomogeneous diffusions

Mihai Gradinaru, Yoann Offret (2013)

Annales de l'I.H.P. Probabilités et statistiques

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Let us consider a solution of a one-dimensional stochastic differential equation driven by a standard Brownian motion with time-inhomogeneous drift coefficient ${ρ sgn (x) | x |}^{α} / t^{β}$ . This process can be viewed as a Brownian motion evolving in a potential, possibly singular, depending on time. We prove results on the existence and uniqueness of solution, study its asymptotic behaviour and made a precise description, in terms of parameters $ρ$ , $α$ and $β$ , of the recurrence, transience and convergence. More precisely,...

Hitting distributions of geometric Brownian motion

T. Byczkowski, M. Ryznar (2006)

Studia Mathematica

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Let τ be the first hitting time of the point 1 by the geometric Brownian motion X(t) = x exp(B(t) - 2μt) with drift μ ≥ 0 starting from x > 1. Here B(t) is the Brownian motion starting from 0 with EB²(t) = 2t. We provide an integral formula for the density function of the stopped exponential functional $A (τ) = \int_{0}^{τ} X ² (t) d t$ and determine its asymptotic behaviour at infinity. Although we basically rely on methods developed in [BGS], the present paper covers the case of arbitrary drifts μ ≥ 0 and provides...

Minkowski sums and Brownian exit times

Christer Borell (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

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If $C$ is a domain in $^{n},$ the Brownian exit time of $C$ is denoted by $T_{C} .$ Given domains $C$ and $D$ in $^{n}$ this paper gives an upper bound of the distribution function of $T_{C + D}$ when the distribution functions of $T_{C}$ and $T_{D}$ are known. The bound is sharp if $C$ and $D$ are parallel affine half-spaces. The paper also exhibits an extension of the Ehrhard inequality

Central limit theorems for the brownian motion on large unitary groups

Florent Benaych-Georges (2011)

Bulletin de la Société Mathématique de France

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In this paper, we are concerned with the large $n$ limit of the distributions of linear combinations of the entries of a Brownian motion on the group of $n \times n$ unitary matrices. We prove that the process of such a linear combination converges to a Gaussian one. Various scales of time and various initial distributions are considered, giving rise to various limit processes, related to the geometric construction of the unitary Brownian motion. As an application, we propose a very short proof of...

Characterizations of processes with stationary and independent increments under $G$ -expectation

Yongsheng Song (2013)

Annales de l'I.H.P. Probabilités et statistiques

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Our purpose is to investigate properties for processes with stationary and independent increments under $G$ -expectation. As applications, we prove the martingale characterization of $G$ -Brownian motion and present a pathwise decomposition theorem for generalized $G$ -Brownian motion.

Upper bounds for the density of solutions to stochastic differential equations driven by fractional brownian motions

Fabrice Baudoin, Cheng Ouyang, Samy Tindel (2014)

Annales de l'I.H.P. Probabilités et statistiques

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In this paper we study upper bounds for the density of solution to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H g t; 1 / 3$ . We show that under some geometric conditions, in the regular case $H g t; 1 / 2$ , the density of the solution satisfies the log-Sobolev inequality, the Gaussian concentration inequality and admits an upper Gaussian bound. In the rough case $H g t; 1 / 3$ and under the same geometric conditions, we show that the density of the solution is smooth and...