Marco Carfagnini, Maria Gordina
A small ball problem and Chung's law of iterated logarithm for a hypoelliptic Brownian motion in Heisenberg group are proven. In addition, bounds on the limit in Chung's law are established.
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This paper gives sufficent and necessary conditions on a kind of limit results to hold on the precise convergent rate of an infinite series of probabilities on the Chung type law of the iterated logarithm.
We prove a new Donsker's invariance principle for independent and identically distributed random variables under the sub-linear expectation. As applications, the small deviations and Chung's law of the iterated logarithm are obtained.
Chung's Law of the Iterated Logarithm for a Class of Stochastic Heat Equations
该研究针对一类系数依赖解的乘性噪声驱动随机热方程，结合小球概率文献与冻结系数技术，建立了其解的Chung型重对数律，且几乎必然可确定其中的极限常数。
We establish a Chung-type law of the iterated logarithm and the exact local and uniform moduli of continuity for a large class of anisotropic Gaussian random fields with a harmonizable-type integral representation and the property of strong local nondeterminism. Compared with the existing results in the literature, our results do not require the assumption of stationary increments and provide more precise upper and lower bounds for the limiting constants. The results are applicable to the solutions of a class of linear stochastic partial differential equations driven by a fractional-colored Ga…
Small ball probabilities and Chung's law of the iterated logarithm for Gaussian Volterra processes with power-type kernels
该研究针对带幂型核的高斯Volterra过程，推导其小球概率双边估计，进而证明各固定点、原点及无穷远处的Chung重对数律，明确参数对过程特性的决定作用。
Khinchin's and Chung's Laws of the Iterated Logarithm at Time Zero for the Linear Stochastic Fractional Diffusion Equation
该研究针对带零初值的线性随机分数阶扩散方程，建立了时间零点的辛钦型重对数律，在特定附加条件下证明了钟氏型重对数律，将随机热方程的初始时间重对数律推广至一类时间分数阶随机扩散方程。
We prove Chung-type laws of the iterated logarithm for general Lévy processes at zero. In particular, we provide tools to translate small deviation estimates directly into laws of the iterated logarithm. This reveals laws of the iterated logarithm for Lévy processes at small times in many concrete examples. In some cases, exotic norming functions are derived.
Consider the first exit time $T_{a,b}$ from a finite interval $[-a,b]$ for an homogeneous fluctuating functional $X$ of a linear Brownian motion. We show the existence of a finite positive constant $\k$ such that $$\lim_{t\to\infty}t^{-1}\log \p[ T_{ab} > t] = -\k.$$ Following Chung's original approach, we deduce a "liminf" law of the iterated logarithm for the two-sided supremum of $X$. This extends and gives a new point of view on a result of Khoshnevisan and Shi.
Chung-type laws of the iterated logarithm for $m$-fold weighted integrated fractional processes
本文研究了m重加权积分分数过程的Chung型迭代对数定律，推导了极限行为并给出了具体常数，同时探讨了随机化play-the-winner规则的迭代对数定律。
We study the almost sure behavior of solutions of stochastic differential equations (SDEs) as time goes to zero. Our main general result establishes a functional law of the iterated logarithm (LIL) that applies in the setting of SDEs with degenerate noise satisfying the weak Hormander condition but not the strong Hormander condition}. That is, SDEs in which the drift terms must be used in order to conclude hypoellipticity. As a corollary of this result, we obtain the almost sure behavior as time goes to zero of a given direction in the equation, even if noise is not present explicitly in that …
The generalized fractional Brownian motion (GFBM) $X:=\{X(t)\}_{t\ge0}$ with parameters $γ\in [0, 1)$ and $α\in \left(-\frac12+\fracγ{2}, \, \frac12+\fracγ{2} \right)$ is a centered Gaussian $H$-self-similar process introduced by Pang and Taqqu (2019) as the scaling limit of power-law shot noise processes, where $H = α-\fracγ{2}+\frac12 \in(0,1)$. When $γ= 0$, $X$ is the ordinary fractional Brownian motion. For $γ\in (0, 1)$, GFBM $X$ does not have stationary increments, and its sample path properties such as Hölder continuity, path differentiability/non-differentiability, and the functional l…
Let $X:=\{X(t)\}_{t\ge0}$ be a generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019): $$ \big\{X(t)\big\}_{t\ge0}\overset{d}{=}\left\{ \int_{\mathbb R} \left((t-u)_+^α-(-u)_+^α \right) |u|^{-γ} B(du) \right\}_{t\ge0}, $$ with parameters $γ\in (0, 1/2)$ and $α\in \left(-\frac12+ γ, \, \frac12+ γ\right)$. Continuing the studies of sample path properties of GFBM $X$ in Ichiba, Pang and Taqqu (2021) and Wang and Xiao (2021), we establish integral criteria for the lower functions of $X$ at $t=0$ and at infinity by modifying the arguments of Talagrand (1996). As a consequ…
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.
We prove a Chung-type law of the iterated logarithm for a multiparameter extension of the fractional Brownian motion which is not increment stationary. This multiparameter fractional Brownian motion behaves very differently at the origin and away from the axes, which also appears in the Hausdorff dimension of its range and in the measure of its pointwise Hölder exponents. A functional version of this Chung-type law is also provided.
We study the small-time asymptotics for hypoelliptic diffusion processes conditioned by their initial and final positions, in a model class of diffusions satisfying a weak Hörmander condition where the diffusivity is constant and the drift is linear. We show that, while the diffusion bridge can exhibit a blow-up behaviour in the small time limit, we can still make sense of suitably rescaled fluctuations which converge weakly. We explicitly describe the limit fluctuation process in terms of quantities associated to the unconditioned diffusion. In the discussion of examples, we also find an expr…
Small-time asymptotics for hypoelliptic diffusions
本文针对具有多项式漂移和退化加性噪声的扩散过程，提出一种归纳程序以计算其小时间渐近行为，通过两种重标度引出控制问题并求解，从而给出边界点正则性的实用判据。
We show some Chung-type $\liminf$ law of the iterated logarithm results at zero for a class of (pure-jump) Feller or Lévy-type processes. This class includes all Lévy processes. The norming function is given in terms of the symbol of the infinitesimal generator of the process. In the Lévy case, the symbol coincides with the characteristic exponent.
Lower classes and Chung's LILs of the fractional integrated generalized fractional Brownian motion
本研究针对Pang和Taqqu提出的广义分数布朗运动的分数阶积分过程，借助Talagrand的论证方法与小球概率估计，得到其在零点和无穷远下类的积分准则及Chung型重对数律，丰富了分数布朗运动相关极限理论。
We introduce a class of iterated processes called $α$-time Brownian motion for $0<α\leq 2$. These are obtained by taking Brownian motion and replacing the time parameter with a symmetric $α$-stable process. We prove a Chung-type law of the iterated logarithm (LIL) for these processes which is a generalization of LIL proved in \cite{hu} for iterated Brownian motion. When $α=1$ it takes the following form $$ \liminf_{T\to\infty}T^{-1/2}(\log \log T) \sup_{0\leq t\leq T}|Z_{t}|=π^{2}\sqrt{λ_{1}} a.s. $$ where $λ_{1}$ is the first eigenvalue for the Cauchy process in the interval $[-1,1].$ We also…
Small-ball constants, and exceptional flat points of SPDEs
本文研究乘性噪声随机热方程在中偏差情形下的小球概率，证明小球常数存在并关联已知量，同时发现使 Chung 型重对数律失效的新异常空间点。
Concentration inequalities for strong laws and laws of the iterated logarithm
本文为独立同分布随机变量和推导集中不等式，得到多个强大数定律的非渐近推广及重对数不等式，并给出路径wise类比。
We establish estimates for the local and uniform moduli of continuity of the local time of multifractional Brownian motion, $B^H=(B^{H(t)}(t),t\in\mathbb{R}^+)$. An analogue of Chung's law of the iterated logarithm is studied for $B^H$ and used to obtain the pointwise Hölder exponent of the local time. A kind of local asymptotic self-similarity is proved to be satisfied by the local time of $B^H$.
Let $B^{H, K}= \big\{B^{H, K}(t), t \in \R_+ \big\}$ be a bifractional Brownian motion in $\R^d$. We prove that $B^{H, K}$ is strongly locally nondeterministic. Applying this property and a stochastic integral representation of $B^{H, K}$, we establish Chung's law of the iterated logarithm for $B^{H, K}$, as well as sharp Hölder conditions and tail probability estimates for the local times of $B^{H, K}$. We also consider the existence and the regularity of the local times of multiparameter bifractional Brownian motion $B^{\bar{H}, \bar{K}}= \big\{B^{\bar{H}, \bar{K}}(t), t \in \R^N_+ \big\}$ i…
Strong Invariance Principle for Random Dirichlet Series
该研究证明了随机素数狄利克雷级数可由布朗运动逼近，得到其相关重对数律，还建立了特定参数下两类随机狄利克雷级数的逼近关系及另一类级数的强不变性原理。
Deviation frequencies of Brownian path property approximations
本研究针对布朗运动的多种经典样本路径性质近似，结合几乎必然收敛速率与收敛模可积性的权衡关系及偏差频率，提出几乎必然收敛的量化方法，覆盖相关核心定理与近似内容。
Sample path properties and small ball probabilities for stochastic fractional diffusion equations
本文研究由分数阶时空高斯噪声驱动的随机时空分数阶扩散方程，证明了解的存在唯一性，并建立了精确的一致与局部连续模、Chung型重对数律及小偏差概率结果。
Sharp regularity and small ball probabilities for the stochastic heat equation on bounded domains
本文研究有界域上带分数阶空间相关噪声的随机热方程，证明温和解存在唯一性的充要条件，并给出最优Holder正则性、连续模、重对数律及尖锐小球概率估计。
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