首页/论文/1104.1972

arXiv 1104.19722011-04-11math.PR

Smooth density for some nilpotent rough differential equations

Yaozhong Hu, Samy Tindel

备注：: Contains a Norris type lemma in the rough paths context. A similar result is to be posted soon by M. Hairer and N. Pillai
发表信息：: Journal of Theoretical Probability 26 (2013) 722-749

英文摘要：

In this note, we provide a non trivial example of differential equation driven by a fractional Brownian motion with Hurst parameter 1/3 < H < 1/2, whose solution admits a smooth density with respect to Lebesgue's measure. The result is obtained through the use of an explicit representation of the solution when the vector fields of the equation are nilpotent, plus a Norris type lemma in the rough paths context.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Central limit theorems for non-symmetric random walks on nilpotent covering graphs: Part I

In the present paper, we study central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a point of view of discrete geometric analysis developed by Kotani and Sunada. We establish a semigroup CLT for a non-symmetric random walk on a nilpotent covering graph. Realizing the nilpotent covering graph into a nilpotent Lie group through a discrete harmonic map, we give a geometric characterization of the limit semigroup on the nilpotent Lie group. More precisely, we show that the limit semigroup is generated by the sub-Laplacian with a non-trivial drift on the n…

2019-03-23math.PR

Central limit theorems for non-symmetric random walks on nilpotent covering graphs: Part II

In the present paper, as a continuation of our preceding paper [10], we study another kind of central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a viewpoint of discrete geometric analysis developed by Kotani and Sunada. We introduce a one-parameter family of random walks which interpolates between the original non-symmetric random walk and the symmetrized one. We first prove a semigroup CLT for the family of random walks by realizing the nilpotent covering graph into a nilpotent Lie group via discrete harmonic maps. The limiting diffusion semigroup i…

2019-01-31math.PR

Smoothing effect of rough differential equations driven by fractional Brownian motions

In this work we study the smoothing effect of rough differential equations driven by a fractional Brownian motion with parameter $H>1/4$. The regularization estimates we obtain generalize to the fractional Brownian motion previous results by Kusuoka and Stroock and can be seen as a quantitative version of the existence of smooth densities under Hörmander's type conditions.

2013-04-17math.PR

Wong-Zakai type approximations of rough random dynamical systems by smooth noise

This paper is devoted to the smooth and stationary Wong-Zakai approximations for a class of rough differential equations driven by a geometric fractional Brownian rough path $\boldsymbolω$ with Hurst index $H\in(\frac{1}{3},\frac{1}{2}]$. We first construct the approximation $\boldsymbolω_δ$ of $\boldsymbolω$ by probabilistic arguments, and then using the rough path theory to obtain the Wong-Zakai approximation for the solution on any finite interval. Finally, both the original system and approximative system generate a continuous random dynamical systems $φ$ and $φ^δ$. As a consequence of the…

2022-10-09math.PR

Euler Estimates of Rough Differential Equations

We consider controlled differential equations and give new estimates for higher order Euler schemes. Our proofs are inspired by recent work of A. M. Davie who considers first and second order schemes. In order to implement the general case we make systematic use of geodesic approximations in the free nilpotent group. As application, we can control moments of solutions to rough path differential equations (RDEs) driven by random rough paths with sufficient integrability and have a criteria for L^q - convergence in the Universal Limit Theorem. We also obtain Azencott type estimates and asymptoti…

2006-02-16math.CAmath.PR

Smoothness of the density for solutions to Gaussian rough differential equations

We consider stochastic differential equations of the form $dY_t=V(Y_t)\,dX_t+V_0(Y_t)\,dt$ driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields $V_0$ and $V=(V_1,\ldots,V_d)$ satisfy Hörmander's bracket condition, we demonstrate that $Y_t$ admits a smooth density for any $t\in(0,T]$, provided the driving noise satisfies certain nondegeneracy assumptions. Our analysis relies on relies on an interplay of rough path theory, Malliavin calculus and the theory of Gaussian processes. Our result applies to a broad range of examples including fractional Brownian m…

2015-01-20math.PR

更多相关论文收起更多论文

Dimer-dimer correlations at the rough-smooth boundary

Three phases of macroscopic domains have been seen for large but finite periodic dimer models; these are known as the frozen, rough and smooth phases. The transition region between the frozen and rough region has received a lot of attention for the last twenty years and recently work has been underway to understand the rough-smooth transition region in the case of the two-periodic Aztec diamond. We compute uniform asymptotics for dimer-dimer correlations of the two-periodic Aztec diamond when the dimers lie in the rough-smooth transition region. These asymptotics rely on a formula found in [5]…

2022-02-01math-phmath.MPmath.PR

Airy process at a thin rough region between frozen and smooth

We show there is a last path at the rough smooth boundary of the two-periodic Aztec diamond with parameter $a\in (0,1)$ that, suitably rescaled, converges to the Airy process, under the condition that $a$ tends to zero as the size of the Aztec diamond tends to infinity at a certain rate. This condition causes the rough region to have a thin, mesoscopic width. We also show that the dimers are described by a discrete Bessel kernel when the width is only of microscopic size.

2023-02-09math-phmath.MPmath.PR

光滑粗糙路径、其几何与代数重整化

Smooth rough paths, their geometry and algebraic renormalization

本文引入光滑粗糙路径，利用Maurer-Cartan视角实现Lyons延拓定理的纯代数形式及粗糙路径重整化，并扩展至拟几何与Hopf代数框架。

2021-11-30math.CAmath.PRmath.RA

Local Geometry of the rough-smooth interface in the two-periodic Aztec diamond

Random tilings of the two-periodic Aztec diamond contain three macroscopic regions: frozen, where the tilings are deterministic; rough, where the correlations between dominoes decay polynomially; smooth, where the correlations between dominoes decay exponentially. In a previous paper, the authors found that a certain averaging of height function differences at the rough-smooth interface converged to the extended Airy kernel point process. In this paper, we augment the local geometrical picture at this interface by introducing well-defined lattice paths which are closely related to the level li…

2021-04-23math-phmath.MPmath.PR

双重粗糙-平滑边界的Airy线集合

The Airy line ensemble at the rough-smooth boundary

Durham University（杜伦大学）；University of Toronto（多伦多大学）

本文研究了双重粗糙-平滑边界在双周期Aztec菱形中的特性，证明了高度函数收敛于Airy表面和噪声场之和，并首次展示了无向路径向Airy线集合的收敛结果。

2026-02-27math-phmath.MPmath.PR

Smooth approximation of Yang--Mills theory on $\mathbb{R}^{2}$: a rough path approach

In the context of rough path theory (RPT), the theories of Hairer (2014) and Gubinelli--Imkeller--Perkowski (2015) (GIP theory) gave new methods for construction of $Φ_{3}^{4}$ model. Roughly, their results state that a quantum field in a $Φ_{3}^{4}$ model can be smoothly approximated. Consider the following question: Can RPT be applied to quantum Yang--Mills (YM) gauge field theories to show that any YM theory can be smoothly approximated? In this paper we consider this problem in the simplest case of Euclidean YM theory, i.e. YM on $\mathbb{R}^{2}$ with the usual Euclidean metric, as a test …

2017-05-06math-phmath.MPmath.PR