在Carnot群上利用Legendre展开的固定时刻布朗运动耦合策略

分数布朗运动的谱表示与模拟

Spectral Representation and Simulation of Fractional Brownian Motion

本文提出一种基于谱表示和勒让德多项式正交基的新方法，用于在连续时间下模拟分数布朗运动，并给出了算法、理论依据及数值实验结果。

Density of the signature process of fBm

We study the density of the signature of fractional Brownian motions with parameter $H>1/4$. In particular, we prove existence, smoothness, global Gaussian upper bounds and Varadhan's type asymptotics for this density. A key result is that the estimates on the density we obtain are controlled by the Carnot-Carathéodory distance of the group.

A semicircle law and decorrelation phenomena for iterated Kolmogorov loops

We consider a standard one-dimensional Brownian motion on the time interval $[0,1]$ conditioned to have vanishing iterated time integrals up to order $N$. We show that the resulting processes can be expressed explicitly in terms of shifted Legendre polynomials and the original Brownian motion, and we use these representations to prove that the processes converge weakly as $N\to\infty$ to the zero process. This gives rise to a polynomial decomposition for Brownian motion. We further study the fluctuation processes obtained through scaling by $\sqrt{N}$ and show that they converge in finite dime…

Large deviations for many Brownian bridges with symmetrised initial-terminal condition

Consider a large system of $N$ Brownian motions in $\mathbb{R}^d$ with some non-degenerate initial measure on some fixed time interval $[0,β]$ with symmetrised initial-terminal condition. That is, for any $i$, the terminal location of the $i$-th motion is affixed to the initial point of the $σ(i)$-th motion, where $σ$ is a uniformly distributed random permutation of $1,...,N$. Such systems play an important role in quantum physics in the description of Boson systems at positive temperature $1/β$. In this paper, we describe the large-N behaviour of the empirical path measure (the mean of the Di…

On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group

In this note, we derive a new logarithmic Sobolev inequality for the heat kernel on the Heisenberg group. The proof is inspired from the historical method of Leonard Gross with the Central Limit Theorem for a random walk. Here the non commutative nature of the increments produces a new gradient which naturally involves a Brownian bridge on the Heisenberg group. This new inequality contains the optimal logarithmic Sobolev inequality for the Gaussian distribution in two dimensions. We compare this new inequality with the sub-elliptic logarithmic Sobolev inequality of Hong-Quan Li and with the mo…

使用概率空间上的变分论证对一类粘性 Hamilton--Jacobi 方程进行均匀化

Homogenization of some viscous Hamilton--Jacobi equations using variational arguments on probability spaces

本文通过概率空间上的变分方法，利用随机控制表示和Gamma-收敛，识别均匀化拉格朗日量并经勒让德变换得到有效哈密顿量，完整刻画粘性Hamilton-Jacobi方程的均匀化极限，并给出对局部缺陷的稳定性结果。

Large deviations for empirical path measures in cycles of integer partitions

Consider a large system of $N$ Brownian motions in $\mathbb{R}^d$ on some fixed time interval $[0,β]$ with symmetrised initial-terminal condition. That is, for any $i$, the terminal location of the $i$-th motion is affixed to the initial point of the $σ(i)$-th motion, where $σ$ is a uniformly distributed random permutation of $1,...,N$. In this paper, we describe the large-N behaviour of the empirical path measure (the mean of the Dirac measures in the $N$ paths) when $ Λ\uparrow\mathbb{R}^d $ and $ N/|Λ|\toρ$. The rate function is given as a variational formula involving a certain entropy fun…