Gaussian lower bound for the FIN diffusion
Manuel Cabezas
我们建立了高斯下界的FIN奇异扩散使用Ray-Knight描述的本地时间。
We establish a Gaussian lower bound for the FIN singular diffusion using the Ray-Knight description of the local time.
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We consider an iterated Kolmogorov diffusion $X_{t}$ of step $n$. The small ball problem for $X_{t}$ is solved by means of the Gaussian correlation inequality. We also prove Chung's laws of iterated logarithm for $X_{t}$ both at time zero and infinity.
Quenched large deviations for diffusions in a random Gaussian shear flow drift
本文证明了具有随机漂移V的扩散过程在大时间内的完全大偏差原理，V是中心高斯剪切流随机场。大偏差原理是在“淬火”条件下建立的，即在V的随机性中几乎肯定是有效的。
We study Gaussian approximations to the distribution of a diffusion. The approximations are easy to compute: they are defined by two simple ordinary differential equations for the mean and the covariance. Time correlations can also be computed via solution of a linear stochastic differential equation. We show, using the Kullback-Leibler divergence, that the approximations are accurate in the small noise regime. An analogous discrete time setting is also studied. The results provide both theoretical support for the use of Gaussian processes in the approximation of diffusions, and methodological…
Fractional diffusion in Gaussian noisy environment
我们研究了高斯噪声环境中的分数阶随机偏方程所描述的分数阶扩散：$D_t^αu(t, x)=\textit{B}u+u\cdot W^H$，其中$D_t^α$是分数阶$α$相对于时间变量$t$的导数，$\textit{B}$是二阶椭圆算子相对于空间变量$x\in\mathbb{R}^d$，$W^H$是分数阶高斯噪声的赫斯特参数$H=(H_1, \cdots, H_d)$。我们得到了$α$和$H$满足的条件，使得平方可积解$u$存在唯一.
We obtain non-asymptotic Gaussian concentration bounds for the difference between the invariant measure $ν$ of an ergodic Brownian diffusion process and the empirical distribution of an approximating scheme with decreasing time step along a suitable class of (smooth enough) test functions f such that f -- $ν$(f) is a coboundary of the infinitesimal generator. We show that these bounds can still be improved when the (squared) Fr{ö}benius norm of the diffusion coefficient lies in this class. We apply these bounds to design computable non-asymptotic confidence intervals for the approximating sche…
Stochastic fractional diffusion equations with spatial Gaussian noise in the Stratonovich regime
针对一类随机分数阶扩散方程，研究人员证明了Stratonovich解的存在性，得到了特定范围p下p阶矩的大t和大p渐近行为对应的Lyapunov指数精确值。
We prove transportation-cost inequalities for the law of SDE solutions driven by general Gaussian processes. Examples include the fractional Brownian motion, but also more general processes like bifractional Brownian motion. In case of multiplicative noise, our main tool is Lyons' rough paths theory. We also give a new proof of Talagrand's transportation-cost inequality on Gaussian Fréchet spaces. We finally show that establishing transportation-cost inequalities implies that there is an easy criterion for proving Gaussian tail estimates for functions defined on that space. This result can be …
In this article, we consider the following stochastic fractional diffusion equation \begin{equation*} \left(\partial^β+\dfracν{2}\left(-Δ\right)^{α/ 2}\right) u(t, x)= λ\: I_{0_+}^γ\left[u(t, x) \dot{W}(t, x)\right] ,\quad t>0,\: x \in \mathbb{R}, \end{equation*} where $α>0$, $β\in(0,2]$, $γ\ge 0$, $λ\neq0$, $ν>0$, and $\dot{W}$ is a Gaussian noise which is white or fractional in time and rough in space. We prove the existence and uniqueness of the solution in the Itô-Skorohod sense and obtain the lower and upper bounds for the $p$-th moment. The Hölder regularity of the solution is also studi…
This paper studies the linear stochastic partial differential equation of fractional orders both in time and space variables $\left(\partial^β+ \fracν{2} (-Δ)^{α/2} \right) u(t,x)= λu(t,x) \dot{W}(t,x)$, where $\dot W$ is a general Gaussian noise and $β\in (1/2, 2)$, $α\in (0, 2]$. The existence and uniqueness of the solution, the moment bounds of the solution are obtained by using the fundamental solutions of the corresponding deterministic counterpart represented by the Fox H-functions. Along the way, we obtain some new properties of the fundamental solutions.
We consider the behavior of the Gaussian concentration bound (GCB) under stochastic time evolution. More precisely, we consider a Markovian diffusion process on $\mathbb{R}^d$ and start the process from an initial distribution $μ$ that satisfies GCB. We then study the question whether GCB is preserved under the time-evolution, and if yes, how the constant behaves as a function of time. In particular, if for the constant we obtain a uniform bound, then we can also conclude properties of the stationary measure(s) of the diffusion process. This question, as well as the methodology developed in th…
Skew-Normal Diffusions
本文提出偏正态扩散（SKN）过程，通过Doob h-变换构造对应随机微分方程，证明其保留高斯过程线性变换不变性，拓展了Kalman-Bucy滤波框架。
Shot noise processes have been extensively studied due to their mathematical properties and their relevance in several applications. Here, we consider nonnegative shot noise processes and prove their weak convergence to Lévy-driven Ornstein-Uhlenbeck (OU), whose features depend on the underlying jump distributions. Among others, we obtain the OU-Gamma and OU-Inverse Gaussian processes, having gamma and inverse gaussian processes as background Lévy processes, respectively. Then, we derive the necessary conditions guaranteeing the diffusion limit to a Gaussian OU process, show that they are not …
Superdiffusive bounds on self-repellent Brownian polymers and diffusion in the curl of the Gaussian free field in d=2
本文考虑二维随机环境中的两个随机扩散模型。第一个例子是自排斥的布朗聚合物，这描述了由其自身的占据时间度量（本地时间）的负梯度推动的扩散。第二个例子是无质量高斯自由场的旋度所给出的固定随机环境中的扩散。在这两种情况下，我们表明，该过程是超扩散：方差增长速度比线性随时间。我们分别给出了t log log t阶的下界和上界，t log t.我们还提出了一个各向异性版本的自排斥布朗聚合物的计算，我们给下和上界的t（log t）^{1/2}，分别，t log t。在拉普拉斯变换的意义下给出了其上界，证明依赖于预解式方法.对于各向同性情况，这些过程的方差的真实阶数预期为t（log t）^{1/2}，对于非各向同性情况，预期为t（log t）^{2/3}。在附录中，我们提出了这些标度指数的非严格推导。
Gaussian integrability of distance function under the Lyapunov condition
本文给出了当对称扩散马氏算子满足李雅普诺夫条件时，距离函数为$μe^{δd^2(x,x_0)} < \infty$对某些$δ>0$的高斯可积性的直接证明，从而回答了Cattiaux-Guillin-Wu [6，Page 295]中提出的一个问题.类似的论点仍然适用于具有无界扩散系数的扩散过程和跳跃过程，如生灭链。在由[9，命题3.5]产生的Gozlan条件下也进行了类似的讨论。
Sub-Gaussian heat kernel estimates for reflected diffusion on inner uniform domains
该研究证明环境空间中扩散过程的次高斯热核估计性质，可被内一致区域子集上的反射扩散过程继承，推进了相关扩散过程热核性质研究。
Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction
我们考虑了一个空间约束下的$N$无序平均场相互作用扩散系统：每个粒子$θ_i$都附着在周期晶格的一个位置$x_i$上，粒子$θ_i$和$θ_j$之间的相互作用随着$| x_i-x_j|^{-α}$（对于$α\in[0,1)$）而减小.在以前的工作中，它被证明在大人口的粒子的经验措施收敛到一个非线性偏微分方程的McKean-Vlasov型的解决方案。本文的目的是研究与这种收敛相关的波动。我们表现出特别是一个相变中的缩放和波动的性质：当$α\in[0,\frac{1}{2})$，波动是高斯的，由一个线性SPDE，缩放$\sqrt{N}$，而波动是确定性的缩放$N^{1-α}$的情况下$α\in(\frac{1}{2},1)$。
In this paper, we develop asymptotic theory for the mixing detection methodology proposed by M. Magdziarz and A. Weron [Physical Review E, 84:051138 (2011)]. The assumptions cover a broad family of Gaussian stochastic processes including fractional Gaussian noise and the fractional Ornstein-Uhlenbeck process. We show that the asymptotic distribution and convergence rates of the detection statistic may be, respectively, Gaussian or non-Gaussian and standard or nonstandard depending on the diffusion exponent. The results pave the way for mixing detection based on a single observed sample path an…
The standard diffusive spreading, characterized by a Gaussian distribution with mean square displacement that grows linearly with time, can break down, for instance, under the presence of correlations and heterogeneity. In this work, we consider the spread of a population of fractional (long-time correlated) Brownian walkers, with time-dependent and heterogeneous diffusivity. We aim to obtain the possible scenarios related to these individual-level features from the observation of the temporal evolution of the population spatial distribution. We develop and discuss the possibility and limitati…
Generalizing Brownian motion (BM), fractional Brownian motion (FBM) is a paradigmatic selfsimilar model for anomalous diffusion. Specifically, varying its Hurst exponent, FBM spans: sub-diffusion, regular diffusion, and super-diffusion. As BM, also FBM is a symmetric and Gaussian process, with a continuous trajectory, and with a stationary velocity. In contrast to BM, FBM is neither a Markov process nor a martingale, and its velocity is correlated. Based on a recent study of selfsimilar Ito diffusions, we explore an alternative selfsimilar model for anomalous diffusion: fractional Ito motion (…
Diffusions and random walks with prescribed sub-Gaussian heat kernel estimates
本文提出了一种新的扩散构造，用于研究具有预设亚高斯热核估计的随机游走和扩散过程。
Alpha-diversity processes and normalized inverse-Gaussian diffusions
无限多中性等位基因模型最近被推广到一类与两参数Poisson-Dirichlet型Gibbs划分相关的扩散过程。本文引入了一类由归一化逆高斯随机概率测度引起的与Gibbs划分的不同子类相关联的无穷维扩散。这种扩散描述了无限多类型的频率的演变以及由α多样性扩散驱动的随时间变化的突变率的动态。作为一个动态的版本，相对于这个框架，吉布斯分区的相应概念，后者是明确地从一个基本的人口模型，并显示出一致的，在一个特殊的情况下，与扩散近似的一个关键的高尔顿-沃森分支过程。一类无穷维过程的特点是在适当的域上的无穷小生成元，并证明是有限多类型的Feller扩散的某一序列的分布极限。此外，通过适当变换的Moran型粒子过程提供离散表示，其中粒子是来自归一化的逆高斯随机概率测度的样本。讨论了极限扩散与双参数模型的关系。
Dynamical systems driven by nonlinear delay SDEs with small noise can exhibit important rare events on long timescales. When there is no delay, classical large deviations theory quantifies rare events such as escapes from metastable fixed points. Near such fixed points, one can approximate nonlinear delay SDEs by linear delay SDEs. Here, we develop a fully explicit large deviations framework for (necessarily Gaussian) processes $X_t$ driven by linear delay SDEs with small diffusion coefficients. Our approach enables fast numerical computation of the action functional controlling rare events fo…
The stochastic solution with Gaussian stationary increments is establihsed for the symmetric space-time fractional diffusion equation when $0 < β< α\le 2$, where $0 < β\le 1$ and $0 < α\le 2$ are the fractional derivation orders in time and space, respectively. This solution is provided by imposing the identity between two probability density functions resulting (i) from a new integral representation formula of the fundamental solution of the symmetric space-time fractional diffusion equation and (ii) from the product of two independent random variables. This is an alternative method with resp…
Diffusion models for Gaussian distributions: Exact solutions and Wasserstein errors
本文针对高斯数据分布，推导扩散模型反向SDE和概率流ODE的解析解，并计算任意数值格式下的精确Wasserstein误差，从而在数据空间中直接监控收敛性。
One-step lowest-variance selection in a Gaussian random-field model motivated by masked diffusion: Total correlation and a square root collision threshold
受掩码离散扩散启发，研究高斯随机场模型单步选择，用局部相关非负分数场表示不确定性，通过距离相关高斯模型衡量所选位置相关性，建立两个互补结果，为理解相关因素对掩码离散扩散中基于置信度一步选择的影响提供基线。
Learning Mixtures of Gaussians Using Diffusion Models
提出基于扩散模型框架的解析算法，通过推导高斯噪声敏感性界限并利用分段多项式回归学习得分函数，实现了高斯混合及低维流形分布的准多项式时间学习。
Transfer Entropy and Directed Information are information-theoretic measures of the directional dependency between stochastic processes. Following the definitions of Schreiber and Massey in discrete time, we define and evaluate these measures for the components of multidimensional Gaussian diffusion processes. When the components are jointly Markov, the Transfer Entropy and Directed Information are both measures of influence according to a simple physical principle. More generally, the effect of other components has to be accounted for, and this can be achieved in more than one way. We propose…
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