José Luís da Silva, Mohamed Erraoui
In this note we prove that the probability measures generated by two generalized grey Brownian motions with different parameters are singular with respect to each other. This result can be interpreted as an extension of the Feldman-Hájek dichotomy of Gaussian measures to a family of non-Gaussian measures
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We show that the uniform norm of generalized grey Brownian motion over the unit interval has an analytic density, excluding the special case of fractional Brownian motion. Our main result is an asymptotic expansion for the small ball probability of generalized grey Brownian motion, which extends to other norms on path space. The decay rate is not exponential but polynomial, of degree two. For the uniform norm and the Hölder norm, we also prove a large deviations estimate.
The generalized grey Brownian motion is a time continuous self-similar with stationary increments stochastic process whose one dimensional distributions are the fundamental solutions of a stretched time fractional differential equation. Moreover, the distribution of the time-changed Brownian motion by an inverse stable process solves the same equation, hence both processes have the same one dimensional distribution. In this paper we show the mutual singularity of the probability measures on the path space which are induced by generalized grey Brownian motion and the time-changed Brownian motio…
In this paper we investigate the class of grey Brownian motions $B_{α,β}$ ($0<α<2$, $0<β\leq1$). We show that grey Brownian motion admits different representations in terms of certain known processes, such as fractional Brownian motion, multivariate elliptical distribution or as a subordination. The weak convergence of the increments of $B_{α,β}$ in $t$, $w$-variables are studied. Using the Berman criterium we show that $B_{α,β}$ admits a $λ$-square integrable local time $L^{B_{α,β}}(\cdot,I)$ almost surely ($λ$ Lebesgue measure). Moreover, we prove that this local time can be weak-approximate…
We prove that the self-intersection local times for generalized grey Brownian motion $B^{β,α}$ in arbitrary dimension $d$ is a well defined object in a suitable distribution space for $dα<2$.
In this paper we investigate the representation of a class of non Gaussian processes, namely generalized grey Brownian motion, in terms of a weighted integral of a stochastic process which is a solution of a certain stochastic differential equation. In particular the underlying process can be seen as a non Gaussian extension of the Ornstein-Uhlenbeck process, hence generalizing the representation results of Muravlev as well as Harms and Stefanovits to the non Gaussian case.
In this paper we investigate the form factors of paths for a class of non Gaussian processes. These processes are characterized in terms of the Mittag-Leffler function. In particular, we obtain a closed analytic form for the form factors, the Debye function, and can study their asymptotic decay.
In the context of non-Gaussian analysis, Schneider [27] introduced grey noise measures, built upon Mittag-Leffler functions; analogously, grey Brownian motion and its generalizations were constructed (see, for example, [25], [6], [7], [8]). In this paper, we construct and study a new non-Gaussian measure, by means of the incomplete-gamma function (exploiting its complete monotonicity). We label this measure Gamma-grey noise and we prove, for it, the existence of Appell system. The related generalized processes, in the infinite dimensional setting, are also defined and, through the use of the R…
In this article, we show that the standard vector-valued generalization of a generalized grey Brownian motion (ggBm) has independent components if and only if it is a fractional Brownian motion. In order to extend ggBm with independent components, we introduce a vector-valued generalized grey Brownian motion (vggBm). The characteristic function of the corresponding measure is introduced as the product of the characteristic functions of the one-dimensional case. We show that for this measure, the Appell system and a calculus of generalized functions or distributions are accessible. We character…
Rough Bergomi turns grey
本文提出用广义灰色布朗运动替代分数布朗运动扩展粗糙 Bergomi 模型，打破对数正态假设，为 SPX/VIX 期权联合校准提供可行方案，并给出半封闭与渐近公式及数值验证。
We establish the singularity with respect to Lebesgue measure as a function of time of the conditional probability that the sum of two one-dimensional Brownian motions will exit from the unit interval before time $t$, given the trajectory of the second Brownian motion up to the same time. On the way of doing so we show that if one solves the one-dimensional heat equation with zero condition on a trajectory of a one-dimensional Brownian motion, which is the lateral boundary, then for each moment of time with probability one the normal derivative of the solution is zero, provided that the diffus…
We study the total mass of a d-dimensional super-Brownian motion as it first exits an increasing sequence of balls. The process of the total mass is a time-inhomogeneous continuous-state branching process, where the increasing radii of the balls are taken as the time parameter. We are able to characterise its time-dependent branching mechanism and show that it converges, as time goes to infinity, towards the branching mechanism of the total mass of a one-dimensional super-Brownian motion as it first crosses above an increasing sequence of levels. Our results allow us to identify the compact su…
We establish the Lifschitz-type singularity around the bottom of the spectrum for the integrated density of states for a class of subordinate Brownian motions in presence of the nonnegative Poissonian random potentials, possibly of infinite range, on the Sierpiński gasket. We also study the long-time behaviour for the corresponding averaged Feynman-Kac functionals.
We study sufficient conditions which ensure that the probability measures generated by two bifractional Brownian motions on an interval [0,1] are singular with respect to each other and sufficient conditions for the probability measures generated by two trifractional Brownian motions on an interval [0,1] are singular with respect to each other.
Generalized random processes related to Hadamard operators and Le Roy measures
本文提出基于Hadamard算子和Le Roy测度的广义随机过程，用于研究分数偏微分方程及非标准扩散特性。
We consider a one-parameter family of Grushin-type singularities on surfaces, and discuss the possible diffusions that extend Brownian motion to the singularity. This gives a quick proof and clear intuition for the fact that heat can only cross the singularity for an intermediate range of the parameter. When crossing is possible and the singularity consists of one point, we give a complete description of these diffusions, and we describe a "best" extension, which respects the isometry group of the surface and also realizes the unique symmetric one-point extension of the Brownian motion, in the…
Cameron--Martin Type Theorem for a Class of non-Gaussian Measures
本文研究广义灰色布朗运动相关非高斯测度的拟不变性，确定其Cameron--Martin空间并给出Radon-Nikodym密度，证明分部积分公式及算子可闭性。
Generalized Wright Analysis: Stochastic and Applications
本文研究广义赖特分析的随机对应问题，定义了扩展已知过程的广义Fox-$H$非高斯非马尔可夫过程，分析其概率密度、协方差等性质并刻画噪声，建立局部时存在性。
We consider solutions of the linear heat equation in $\mathbb{R}^N$ with isolated singularities. It is assumed that the position of a singular point depends on time and is Hölder continuous with the exponent $α\in (0,1)$. We show that any isolated singularity is removable if it is weaker than a certain order depending on $α$. We also show the optimality of the removability condition by showing the existence of a solution with a nonremovable singularity. These results are applied to the case where the singular point behaves like a fractional Brownian motion with the Hurst exponent $H \in (0,1/2…
Singularity of the loops within a cable-graph loop-soup conditioned by its occupation time
本文研究了在电缆图上布朗运动循环汤在总占用时间场条件下的奇异性质，揭示了循环分布的奇异行为及其成因。
Green Measures for a Class of non-Markov Processes
本文研究广义灰色布朗运动族 $B_{β,α}$ 的格林测度，证明当 $dα>2$ 且 $1<α\le2$ 时永久积分以概率1存在，从而统一推广了分数布朗运动、布朗运动及其他非高斯过程的格林测度。
We extend earlier results on conditioning of super-Brownian motion to general branching rules. We obtain representations of the conditioned process, both as an $h$-transform, and as an unconditioned superprocess with immigration along a branching tree. Unlike the finite-variance branching setting, these trees are no longer binary, and strictly positive mass can be created at branch points. This construction is singular in the case of stable branching. We analyze this singularity first by approaching the stable branching function via analytic approximations. In this context the singularity of t…
We give an elementary self-contained proof of the fact that the walk dimension of the Brownian motion on an arbitrary generalized Sierpiński carpet is greater than two, no proof of which in this generality had been available in the literature. Our proof is based solely on the self-similarity and hypercubic symmetry of the associated Dirichlet form and on several very basic pieces of functional analysis and the theory of regular symmetric Dirichlet forms. We also present an application of this fact to the singularity of the energy measures with respect to the canonical self-similar measure (uni…
We consider the classical problem of existence, uniqueness and asymptotics of monotone solutions to the travelling wave equation associated to the parabolic semi-group equation of a super-Brownian motion with a general branching mechanism. Whilst we are strongly guided by the probabilistic reasoning of Kyprianou (2004) for branching Brownian motion, the current paper offers a number of new insights. Our analysis incorporates the role of Seneta-Heyde norming which, in the current setting, draws on classical work of Grey (1974). We give a pathwise explanation of Evans' immortal particle picture …
This paper studies the weak and strong solutions to the stochastic differential equation $ dX(t)=-\frac12 \dot W(X(t))dt+d\mathcal{B}(t)$, where $(\mathcal{B}(t), t\ge 0)$ is a standard Brownian motion and $W(x)$ is a two sided Brownian motion, independent of $\mathcal{B}$. It is shown that the Itô-McKean representation associated with any Brownian motion (independent of $W$) is a weak solution to the above equation. It is also shown that there exists a unique strong solution to the equation. Itô calculus for the solution is developed. For dealing with the singularity of drift term $\int_0^T \…
Let $M_γ$ be a subcritical Gaussian multiplicative chaos measure associated with a general log-correlated Gaussian field defined on a bounded domain $D \subset \mathbb{R}^d$, $d \geq 1$. We find an explicit formula for its singularity spectrum by showing that $M_γ$ satisfies almost surely the multifractal formalism, i.e., we prove that its singularity spectrum is almost surely equal to the Legendre-Fenchel transform of its $L^q$-spectrum. Then, applying this result, we compute the lower singularity spectrum of the multifractal random walk and of the Liouville Brownian motion.
In this paper we present a general mathematical construction that allows us to define a parametric class of $H$-sssi stochastic processes (self-similar with stationary increments), which have marginal probability density function that evolves in time according to a partial integro-differential equation of fractional type. This construction is based on the theory of finite measures on functional spaces. Since the variance evolves in time as a power function, these $H$-sssi processes naturally provide models for slow and fast anomalous diffusion. Such a class includes, as particular cases, fract…
Removability of the Fundamental Singularity for the Heat Equation and its Consequences. I. Kolmogorov-Petrovsky-type test
本文给出热方程基本奇点可去性的充要条件，等价于奇异Dirichlet问题唯一可解性，并联系抛物测度与条件布朗运动渐近律。
Distinguishing sets and measures from their planar Brownian images
受Hochman问题启发，研究从单个平面布朗像区分确定性时间测度或集合的方法，利用重标相关积分给出像测度律相互奇异性判据，可区分特定自相似集合及其测度，维数足够小时可区分所有不同参数。
Wiener-type Criterion for the Removability of the Fundamental Singularity for the Heat Equation and its Consequences
本文提出新的Weiner型准则，用于任意开集在奇点处的细拓扑薄度条件下对热方程基本奇点可移除性的几何刻画。
We consider a general class of integro-differential evolution equations which includes the governing equation of the generalized grey Brownian motion and the time- and space-fractional heat equation. We present a general relation between the parameters of the equation and the distribution of the underlying stochastic processes, as well as discuss different classes of processes providing stochastic solutions of these equations. For a subclass of evolution equations, containing Saigo-Maeda generalized time-fractional operators, we determine the parameters of the corresponding processes explicitl…
Singularity of Furstenberg measure for infinite covolume discrete subroups in higher rank
本文研究了具有 Property (T) 的半单 Lie 群中无限余体积离散子群上的对称随机游走，证明了此时 Furstenberg 边界上的命中测度具有奇异性，并指出这是高秩情形下的首个此类结果。
Quantitative Propagation of Chaos for Singular Interacting Particle Systems Driven by Fractional Brownian Motion
本文研究了由分数布朗运动驱动的奇异相互作用粒子系统，证明了在粒子数趋于无穷时传播混沌和均场收敛的定量速率。
A class of infinite-dimensional Gaussian processes defined through generalized fractional operators
本文利用由 Bernstein 函数定义的广义分数阶导数与积分扩展无穷维分数阶 Brown 运动构造，分析过程性质与短长程依赖，并给出相应噪声及 Ornstein-Uhlenbeck 型过程。
How to Study Reflected Brownian Motion in a Quadrant via Kernel Functional Equations? A short survey
本文综述了通过核函数方程研究象限中半鞅反射布朗运动的方法，涵盖非退化和退化情形，重点讨论了不变测度、格林函数、调和函数及其拉普拉斯变换的显式表示和渐近分析。
Finite-time blowup of a Brownian particle in a repulsive potential
本文研究过阻尼布朗粒子在快于二次增长的幂律排斥势中的有限时间逃逸，确定平均爆破时间并揭示其概率分布的普适尾部特征。
We analyse q-functional equations arising from tree-like combinatorial structures, which are counted by size, internal path length, and certain generalisations thereof. The corresponding counting parameters are labelled by a positive integer k. We show the existence of a joint limit distribution for these parameters in the limit of infinite size, if the size generating function has a square root as dominant singularity. The limit distribution coincides with that of integrals of k-th powers of the standard Brownian excursion. Our approach yields a recursion for the moments of the limit distribu…
This paper presents a new approach to the analysis of mixed processes \[X_t=B_t+G_t,\qquad t\in[0,T],\] where $B_t$ is a Brownian motion and $G_t$ is an independent centered Gaussian process. We obtain a new canonical innovation representation of $X$, using linear filtering theory. When the kernel \[K(s,t)=\frac{\partial^2}{\partial s\,\partial t}\mathbb{E}G_tG_s,\qquad s\ne t\] has a weak singularity on the diagonal, our results generalize the classical innovation formulas beyond the square integrable setting. For kernels with stronger singularity, our approach is applicable to processes with…
An extension of the stochastic sewing lemma and applications to fractional stochastic calculus
本文推广了 Lê 的随机缝补引理，将条件期望从给定 $\mathcal F_s$ 放宽到给定 $\mathcal F_v$（$v<s$）并引入奇异性以利用渐近去相关性质，从而在分数布朗运动驱动的随机积分近似、局部时表示及随机微分方程强存在唯一性三个方面获得新应用。
In this paper we study a parametric class of stochastic processes to model both fast and slow anomalous diffusion. This class, called generalized grey Brownian motion (ggBm), is made up off self-similar with stationary increments processes (H-sssi) and depends on two real parameters alpha in (0,2) and beta in (0,1]. It includes fractional Brownian motion when alpha in (0,2) and beta=1, and time-fractional diffusion stochastic processes when alpha=beta in (0,1). The latters have marginal probability density function governed by time-fractional diffusion equations of order beta. The ggBm is defi…
IDS for subordinate Brownian motions in Poisson random environment on nested fractals
本文研究了在嵌套分形上带有泊松随机环境的从属布朗运动的IDS奇异性，通过将随机势能转化为合金型势能，拓展了伯恩斯坦函数的应用范围。
Quadratic variations of rough generalized Hermite processes: hidden long memory and a universal Gaussian boundary field
本文研究分数阶滤波广义Hermite过程的二次变差，证明其归一化极限由可见粗糙度决定，而记忆由潜在奇异指数决定，并揭示普适高斯边界场及各向同性极限行为。
Short-time blowup statistics of a Brownian particle in repulsive potentials
研究过阻尼布朗粒子在排斥势中的有限时间爆炸，通过WKB近似计算爆炸时间概率分布短时尾部的前置指数因子，对偶数n仅需WKB解，奇数n需补充内部解并匹配。
First-passage area distribution and optimal fluctuations of fractional Brownian motion
研究分数布朗运动首达面积的概率分布，推导出精确标度关系，确定其大A尾部为肥尾且均值发散，并通过蒙特卡洛模拟验证了大小尾部及最优路径。
Fractional Brownian motion in confining potentials: non-equilibrium distribution tails and optimal fluctuations
本文利用最优涨落方法研究限制势中分数布朗运动的非平衡稳态分布尾部，解析确定尾部指数因子，并开发数值算法处理奇异路径，揭示非平衡特性与时间可逆性破坏。
We present three sets of results for the stationary distribution of a two-dimensional semimartingale reflecting Brownian motion (SRBM) that lives in the nonnegative quadrant. The SRBM data can equivalently be specified by three geometric objects, an ellipse and two lines, in the two-dimensional Euclidean space. First, we revisit the variational problem (VP) associated with the SRBM. Building on Avram, Dai and Hasenbein (2001), we show that the value of the VP at a point in the quadrant is equal to the optimal value of a linear function over a convex domain. Depending on the location of the poi…
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