Yves Le Jan, Olivier Raimond
Using the Wiener chaos decomposition, we show that strong solutions of non Lipschitzian S.D.E.'s are given by random Markovian kernels. The example of Sobolev flows is studied in some detail, exhibiting interesting phase transitions.
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Stochastic integration w.r.t. fractional Brownian motion (fBm) has raised strong interest in recent years, motivated in particular by applications in finance and Internet traffic modelling. Since fBm is not a semi-martingale, stochastic integration requires specific developments. Multifractional Brownian motion (mBm) is a Gaussian process that generalizes fBm by letting the local Hölder exponent vary in time. This is useful in various areas, including financial modelling and biomedicine. In this work we start from the fact, established in \cite[Thm 2.1.(i)]{fBm_to_mBm_HerbinLebovitsVehel}, tha…
Stochastic integration with respect to local time of the Brownian sheet and regularising properties of Brownian sheet paths
本研究将随机局部时空积分推广到布朗单情形，建立对应微积分体系，证明广义双参数Itô公式并推导Davie型不等式，为平均型算子的正则性界估计提供工具。
By constructing a new family of successful couplings, the Driver-type integration by parts formula is established for the operator associated with stochastic differential equation driven by fractional Brownian motion. As applications, shift Harnack type inequalities are presented and then the absolute continuity of the solution is proved.
For a parameter 0<q<1, we use the Jackson q-integral to define integration with respect to the so called q-Brownian motion. Our main results are the q-analogs of the L_2-isometry and of the Ito formula for polynomial integrands. We also indicate how the L_2-isometry extends the integral to more general functions.
This paper generalizes the integration theory for volatility modulated Brownian-driven Volterra processes onto the space G* of Potthoff-Timpel distributions. Sufficient conditions for integrability of generalized processes are given, regularity results and properties of the integral are discussed. We introduce a new volatility modulation method through the Wick product and discuss its relation to the pointwise-multiplied volatility model.
Combinatorial formulas for the moments of the Brownian motion on classical compact Lie groups are obtained. These expressions are deformations of formulas of B. Collins and P. Śniady for moments of the Haar measure and yield a proof of the First Fundamental Theorem of invariant theory and of classical Schur-Weyl dualities based on stochastic calculus.
We prove an infinite dimensional integration by parts formula on the law of the modulus of the Brownian bridge $BB=(BB_t)_{0 \leq t \leq 1}$ from $0$ to $0$ in use of methods from white noise analysis and Dirichlet form theory. Additionally to the usual drift term, this formula contains a distribution which is constructed in the space of Hida distributions by means of a Wick product with Donsker's delta (which correlates with the local time of $|BB|$ at zero). This additional distribution corresponds to the reflection at zero caused by the modulus.
Hadamard fractional Brownian motion: path properties and Wiener integration
本文研究Hadamard分数布朗运动的轨迹性质（如Hölder连续性、拟螺旋行为、幂变差和局部非确定性），并以此为基础发展Wiener积分理论，利用乘法Sonine对刻画再生核Hilbert空间，最终建立重对数律。
We study the issue of integration with respect to the non-commutative fractional Brownian motion, that is the analog of the standard fractional Brownian in a non-commutative probability setting.When the Hurst index $H$ of the process is stricly larger than $1/2$, integration can be handled through the so-called Young procedure. The situation where $H=1/2$ corresponds to the specific free case, for which an It{ô}-type approach is known to be possible.When $H<1/2$, rough-path-type techniques must come into the picture, which, from a theoretical point of view, involves the use of some a-priori-de…
We prove an integration by parts formula on the law of the reflecting Brownian motion $X:=|B|$ in the positive half line, where $B$ is a standard Brownian motion. In other terms, we consider a perturbation of $X$ of the form $X^ε= X+εh$ with $h$ smooth deterministic function and $ε>0$ and we differentiate the law of $X^ε$ at $ε=0$. This infinitesimal perturbation changes drastically the set of zeros of $X$ for any $ε>0$. As a consequence, the formula we obtain contains an infinite dimensional generalized functional in the sense of Schwartz, defined in terms of Hida's renormalization of the squ…
On the infinite dimensional space $E$ of continuous paths from $[0,1]$ to $\mathbb R^n$, $n \ge 3$, endowed with the Wiener measure $μ$, we construct a surface measure defined on level sets of the $L^2$-norm of $n$-dimensional processes that are solutions to a class of stochastic gradient system-type equations, and provide an integration by parts formula involving this surface measure. We follow the approach to surface measures in Gaussian spaces proposed via techniques of Malliavin calculus by Airault and Malliavin in 1988.
Forward stochastic integration for adapted processes w.r.t. Riemann-Liouville fractional Brownian motion (Full version)
本文研究了针对Riemann-Liouville分数布朗运动的前向随机积分的$L^2$-鞅表示及其精确等距性。
For every $d\geq 1$, we consider the $d$-dimensional Hermitian fractional Brownian motion (HfBm), that is the process with values in the space of $(d\times d)$-Hermitian matrices and with upper-diagonal entries given by complex fractional Brownian motions of Hurst index $H\in (0,1)$. We follow the approach of [A. Deya and R. Schott: On the rough paths approach to non-commutative stochastic calculus, JFA (2013)] to define a natural integral with respect to the HfBm when $H>\frac13$, and identify this interpretation with the rough integral with respect to the $d^2$ entries of the matrix. Using t…
We pursue our investigations, initiated in [8], about stochastic integration with respect to the non-commutative fractional Brownian motion (NC-fBm). Our main objective in this paper is to compare the pathwise constructions of [8] with a Skorohod-type interpretation of the integral. As a first step, we provide details on the basic tools and properties associated with non-commutative Malliavin calculus, by mimicking the presentation of Nualart's celebrated treatise [14]. Then we check that, just as in the classical (commutative) situation, Skorohod integration can indeed be considered in the pr…
In this paper, we show that the integration of a stochastic differential equations driven by G-Brownian motion in R can be reduced to the integration of an ordinary differential equations parametrized by a variable in (Ω,F). We study the sample solutions of G-SDEs by an extention of G-Itô formula. And then we also get a comparison theorem for G-SDEs and its applications.
We prove an integration by parts formula for the probability measure induced by the semi-classical Riemmanian Brownian bridge over a manifold with a pole.
Invariant subspaces of Brownian shifts on vector-valued Hardy spaces
本文研究了向量值Hardy空间上Brownian移位的不变子空间，刻画了其结构并解决了相应的酉等价问题。
We provide a new methodology to simulate the first exit times of a vector of Brownian motions from an orthant. This new approach can be used to simulate the first exit times of dimension higher than two. When at least one Brownian motion has non-zero drift, the joint density function of the first exit times in N dimensions needs to be known, or approximated. However, when the drifts are all zero, a simpler simulation algorithm is obtained without using the joint density function.
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…
Random rotational invariance of integration by parts formulas within a Bismut-type approach
该研究在随机微分方程李对称理论框架下，证明Bismut型Malliavin演算启发的分部积分公式的随机旋转不变性，分析驱动布朗运动旋转不变性的作用，并通过二维布朗运动驱动的显式模型验证该性质。
We show that for $γ<\sqrt{4/3}$, it is possible to define the Levy area of a planar Brownian motion with the Liouville measure of intermittency parameter $γ$ as the underlying area measure. We also consider the case of smoother curves, and study some properties of the integration map thus defined.
In this paper, we develop a Young integration theory in dimension 2 which will allow us to solve a non-linear one dimensional wave equation driven by an arbitrary signal whose rectangular increments satisfy some Hölder regularity conditions, for some Hölder exponent greater than 1/2. This result will be applied to the infinite dimensional fractional Brownian motion.
Large time behaviour for the semigroup of the kinetic Brownian motion in the plane
本文研究平面中动能布朗运动半群在大时间下的行为，通过构造布朗运动和独立高斯随机变量，利用马尔可夫对偶计算得到梯度估计及反Poincaré不等式，进而推导出生成元的Liouville性质。
Brownian Motion in a Vector Space over a Local Field is a Scaling Limit
本文证明局部域上向量空间中的 Vladimirov-Taibleson 算子所生成的布朗运动是离散群上随机游走的缩放极限，从而推广了此前仅针对 p-进数的结果，深化了与欧氏空间布朗运动的类比。
We describe the class of functions $f: R^n\to R^m$ which transform a vector Brownian Motion into a martingale and use this description to give martingale characterization of the general measurable solution of the multidimensional Cauchy functional equation.
Higher order integration by parts formulae for Wiener measures on a path space between two curves
本文研究两曲线间路径空间上 Wiener 测度的高阶分部积分，借助随机游走逼近构造 Brownian excursion 与 Brownian house-moving，并用 Symmetrization 简化边界项及给出概率解释。
In this article we consider the KPZ fixed point starting from a two-sided Brownian motion with an arbitrary diffusion coefficient. We apply the integration by parts formula from Malliavin calculus to establish a key relation between the two-point (covariance) function of the spatial derivative process and the location of the maximum of an Airy process plus Brownian motion minus a parabola. Integration by parts also allows us to deduce the density of this location in terms of the second derivative of the variance of the KPZ fixed point. In the stationary regime, we find the same density related…
We study interacting systems of linear Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. Our main objective has been to study the long range behavior of the spacings between the Brownian motions arranged in increasing order. For finitely many Brownian motions interacting in this manner, we characterize drifts for which the family of laws of the vector of spacings is tight, and show its convergence to a unique stationary joint distribution given by independent exponential distributions with varying means. We als…
In this paper we construct a theory of stochastic integration of processes with values in $\mathcal{L}(H,E)$, where $H$ is a separable Hilbert space and $E$ is a UMD Banach space (i.e., a space in which martingale differences are unconditional). The integrator is an $H$-cylindrical Brownian motion. Our approach is based on a two-sided $L^p$-decoupling inequality for UMD spaces due to Garling, which is combined with the theory of stochastic integration of $\mathcal{L}(H,E)$-valued functions introduced recently by two of the authors. We obtain various characterizations of the stochastic integral…
We show an It\^ o's formula for nondegenerate Brownian martingales $X_t=\int_0^t u_s dW_s$ and functions $F(x,t)$ with locally integrable derivatives in $t$ and $x$. We prove that one can express the additional term in Itô's s formula as an integral over space and time with respect to local time.
In a previous work, we proved strong convergence with order $1$ of the Ninomiya-Victoir scheme $X^{NV}$ with time step $T/N$ to the solution $X$ of the limiting SDE when the Brownian vector fields commute. In this paper, we prove that the normalized error process $N \left(X - X^{NV}\right)$ converges to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields and the drift vector field. This result ensures that the strong convergence rate is actually $1$ when the Brownian vector fields commute, but at least one of them does not commute with the drift vector…
We prove an almost sure invariance principle (approximation by d-dimensional Brownian motion) for vector-valued Holder observables of large classes of nonuniformly hyperbolic dynamical systems. These systems include Axiom~A diffeomorphisms and flows as well as systems modelled by Young towers with moderate tail decay rates. In particular, the position variable of the planar periodic Lorentz gas with finite horizon approximates a 2-dimensional Brownian motion.
We study the stationary reflected Brownian motion in a non-convex wedge, which, compared to its convex analogue model, has been much rarely analyzed in the probabilistic literature. We prove that its stationary distribution can be found by solving a two dimensional vector boundary value problem (BVP) on a single curve for the associated Laplace transforms. The reduction to this kind of vector BVP seems to be new in the literature. As a matter of comparison, one single boundary condition is sufficient in the convex case. When the parameters of the model (drift, reflection angles and covariance …
In this paper we study the forward integral of operator-valued processes with respect to a cylindrical Brownian motion. In particular, we provide conditions under which the approximating sequence of processes of the forward integral, converges to the stochastic integral process with respect to Sobolev norms of smoothness alpha < 1/2. This result will be used to derive a new integration by parts formula for the forward integral.
Integration by parts formulas and Lie's symmetries of SDEs
本文利用李对称理论研究自治随机微分方程，建立了强准不变原理与有限维分部积分公式，并应用于布朗运动驱动的随机模型。
We solve infinite-dimensional stochastic differential equations (ISDEs) describing an infinite number of Brownian particles interacting via two-dimensional Coulomb potentials. The equilibrium states of the associated unlabeled stochastic dynamics are the Ginibre random point field and Dyson's measures, which appear in random matrix theory. To solve the ISDEs we establish an integration by parts formula for these measures. Because the long-range effect of two-dimensional Coulomb potentials is quite strong, the properties of Brownian particles interacting with two-dimensional Coulomb potentials …
We extend to the vector-valued situation some earlier work of Ciesielski and Roynette on the Besov regularity of the paths of the classical Brownian motion. We also consider a Brownian motion as a Besov space valued random variable. It turns out that a Brownian motion, in this interpretation, is a Gaussian random variable with some pathological properties. We prove estimates for the first moment of the Besov norm of a Brownian motion. To obtain such results we estimate expressions of the form $\E \sup_{n\geq 1}\|ξ_n\|$, where the $ξ_n$ are independent centered Gaussian random variables with va…
We construct obliquely reflected Brownian motions in all bounded simply connected planar domains, including non-smooth domains, with general reflection vector fields on the boundary. Conformal mappings and excursion theory are our main technical tools. A key intermediate step, which may be of independent interest, is an alternative characterization of reflected Brownian motions in smooth bounded planar domains with a given field of angles of oblique reflection on the boundary in terms of a pair of quantities, namely an integrable positive harmonic function, which represents the stationary dist…
The paper obtains the general form of the cross-covariance function of vector fractional Brownian motion with correlated components having different self-similarity indices.
Operator fractional Brownian fields (OFBFs) are Gaussian, stationary-increment vector random fields that satisfy the operator self-similarity relation {X(c^{E}t)}_{t in R^m} L= {c^{H}X(t)}_{t in R^m}. We establish a general harmonizable representation (Fourier domain stochastic integral) for OFBFs. Under additional assumptions, we also show how the harmonizable representation can be reexpressed as a moving average stochastic integral, thus answering an open problem described in Bierme et al.(2007), "Operator scaling stable random fields", Stochastic Processes and their Applications 117, 312--3…
We consider rough paths with jumps. In particular, the analogue of Lyons' extension theorem and rough integration are established in a jump setting, offering a pathwise view on stochastic integration against cadlag processes. A class of Levy rough paths is introduced and characterized by a sub-ellipticity condition on the left-invariant diffusion vector fields and and a certain integrability property of the Carnot--Caratheodory norm with respect to the Levy measure on the group, using Hunt's framework of Lie group valued Levy processes. Examples of Levy rough paths include standard multi-dimen…
A regularity property of fractional Brownian sheets
该研究引入强可荷函数的定义，证明d≥2维布朗单样本路径几乎必然不具该性质，而分数布朗单在赫斯特参数均值大于(d-1)/d时几乎必然具有该正则性。
In this paper we propose a lift of vector field $X$ on a Riemannian manifold $M$ to a vector field $\tilde{X}$ on the curved Cameron-Martin space $H\left(M\right)$ named orthogonal lift. The construction of this lift is based on a least square spirit with respect to a metric on $H(M)$ reflecting the damping effect of Ricci curvature. Its stochastic extension gives rise to a non-adapted Cameron-Martin vector field on $W_o(M)$. In particular, if $M=\mathbb{R}^d$ with Euclidean metric, then the damp disappears and the lift reduces to the well-known Malliavin's lift. We establish an integration by…
This appendix provides a short proof for sample path continuity of the Brownian motion induced by an arbitrary centered Gaussian measure on a separable Banach space, and also some perturbation results for the spectrum of compact self-adjoint operators on a Hilbert space.
Multidimensional Brownian risk models with random trend
本文研究具有随机趋势的多维布朗风险模型，推导了在随机向量与常数限制下，多维过程超过特定阈值的概率在 \\(u\to\infty\\) 时的渐近表达式。
The paper suggests a way of stochastic integration of random integrands with respect to fractional Brownian motion with the Hurst parameter H> 1/2. The integral is defined initially on the processes that are "piecewise" predictable on a short horizon. Then the integral is extended on a wide class of square integrable adapted random processes. This class is described via a mild restriction on the growth rate of the conditional mean square error for the forecast on an arbitrarily short horizon given current observations; differentiability of Hölder property of any kind or degree is not required …
We establish universality for the largest singular values of products of random matrices with right unitarily invariant distributions, in a regime where the number of matrix factors and size of the matrices tend to infinity simultaneously. The behavior of the largest log singular values coincides with the large N limit of Dyson Brownian motion with a characteristic drift vector consisting of equally spaced coordinates, which matches the large N limit of the largest log singular values of Brownian motion on GL(N, C). Our method utilizes the formalism of multivariate Bessel generating functions,…
Large deviations of slow-fast systems driven by fractional Brownian motion
本文研究慢分量受分数布朗运动扰动、快分量由布朗运动驱动的多尺度随机系统，利用Young积分与分数阶微积分建立均匀化极限下的大偏差原理，并揭示速率函数在H=1/2处的不连续性。
We study a regularization by noise phenomenon for the continuous parabolic Anderson model with a potential shifted along paths of fractional Brownian motion. We demonstrate that provided the Hurst parameter is chosen sufficiently small, this shift allows to establish well-posedness and stability to the corresponding problem - without the need of renormalization - in any dimension. We moreover provide a robustified Feynman-Kac type formula for the unique solution to the regularized problem building upon regularity estimates for the local time of fractional Brownian motion as well as non-linear …
Motivated by applications to quantum field theory we consider Gibbs measures for which the reference measure is Wiener measure and the interaction is given by a double stochastic integral and a pinning external potential. In order properly to characterize these measures through DLR equations, we are led to lift Wiener measure and other objects to a space of configurations where the basic observables are not only the position of the particle at all times but also the work done by test vector fields. We prove existence and basic properties of such Gibbs measures in the small coupling regime by m…
These are lecture notes from a course given at the CRM in Montreal in 1992. They survey the author's attempts to find and understand canonical probabilistic entities in a local field (e.g. p-adic) setting. We propose answers to the related questions: ``What are the analogues for Gaussian measures?'' and ``What are the analogues for Brownian motion and its multiparameter relatives?'' There is a suitable concept of orthogonality in the local field setting, and one can mimic a classical abstract definition of Euclidean Gaussian measures. The resulting theory is, in some sense, an L^\infty rather …
Spectral Simplicity and Joint Eigenvalue Densities for a Non-Gaussian Brownian Time Change
该研究针对单位正方形上带独立非高斯系数狄利克雷特征函数展开构造的速度测度的布朗时间变换，通过鞅论证、酉变换等方法证明其谱简单性并得到联合特征值密度。
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