Michael Scheutzow
We review several competing chaining methods to estimate the supremum, the diameter of the range or the modulus of continuity of a stochastic process in terms of tail bounds of their two-dimensional distributions. Then we show how they can be applied to obtain upper bounds for the growth of bounded sets under the action of a stochastic flow.
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On Talagrand's functional and generic chaining
本文在一般分布情形下构造了 Talagrand 型泛函，利用 generic chaining 方法获得了随机过程所有 $p$ 阶矩上确界的上界，并应用于推导 Johnson-Lindenstrauss 引理、2 阶 Gaussian chaos 矩上界及凸信号恢复。
We modify Talagrand's generic chaining method to obtain upper bounds for all p-th moments of the supremum of a stochastic process. These bounds lead to an estimate for the upper tail of the supremum with optimal deviation parameters. We apply our procedure to improve and extend some known deviation inequalities for suprema of unbounded empirical processes and chaos processes. As an application we give a significantly simplified proof of the restricted isometry property of the subsampled discrete Fourier transform.
As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a {\em stochastic maximal inequality} derived by using the formula for integration by parts and on a new concept named {\em strict countability}, is presented. The main results are some weakconvergence theorems for sequences of separable random fields of discrete-time martingales under the uniform topology with the help also of entropy methods. As special cases, some new results for i.i.d.\ random sequences, including a new Donsker t…
The first step to study lower bounds for a stochastic process is to prove a special property - Sudakov minoration. The property means that if a certain number of points from the index set are well separated then we can provide an optimal type lower bound for the mean value of the supremum of the process. Together with the generic chaining argument the property can be used to fully characterize the mean value of the supremum of the stochastic process. In this article we prove the property for canonical processes based on radial-type log concave measures.
We show that a stochastic flow which is generated by a stochastic differential equation on $\R^d$ with bounded volatility has a random attractor provided that the drift component in the direction towards the origin is larger than a certain strictly positive constant $β$ outside a large ball. Using a similar approach, we provide a lower bound for the linear growth rate of the inner radius of the image of a large ball under a stochastic flow in case the drift component in the direction away from the origin is larger than a certain strictly positive constant $β$ outside a large ball. To prove the…
Sharp supremum and Hölder bounds for stochastic integrals indexed by a parameter
本文利用Kolmogorov型链式论证，为2-光滑Banach空间中随机卷积的上确界及参数化随机积分的连续模提供了精确界，建立了任意有界子集上的Hölder空间随机积分理论，并应用于Ornstein-Uhlenbeck过程的长期界和抛物Anderson模型的连续模研究。
We determine the Lyapunov spectrum and stable manifolds of some stochastic flows on the Poincaré group associated to Dudley's relativistic processes.
Strong measurable continuous modifications of stochastic flows
本文提出一种新构造方法，在局部紧可分度量空间中为度量图的可测映射随机流生成强可测连续修正。
In this article we consider multiplicative operator-valued white noise functionals related to a stochastic flow. A generalization of the Krylov-Veretennikov expansion is presented. An analog of such expansion for the Arratia flow is derived.
In this paper we consider the asymptotic behaviour of all moments of the interparticle distance and of all mixed moments of an isotropic Brownian stochastic flow which serves as a smooth approximation of the Arratia flow.
Maximal Solutions and Stochastic Free Boundary Formulations for Stochastic Willmore and Surface Diffusion Flows on $\R^2$
本文研究了随机Willmore流和表面扩散流在R²上的最大解问题，通过将问题转化为随机Stefan问题并应用相关理论，证明了局部强解的存在性和唯一性。
In this article we prove that stochastic differential equation (SDE) with Sobolev drift on compact Riemannian manifold admits a unique $ν$-almost everywhere stochastic invertible flow, where $ν$ is the Riemannian measure, which is quasi-invariant with respect to $ν$. In particular, we extend the well known DiPerna-Lions flows of ODEs to SDEs on Riemannian manifold.
We show how the theory of stochastic flows allows to recover in an elementary way a well known result of Warren on the sticky Brownian motion equation.
Continuous stochastic flows driven by white noise and their duals
研究一类由时空白噪声驱动的连续随机流，通过显式随机微分方程刻画其对偶流，并证明系数近似下解的收敛性，应用于平方贝塞尔流、雅可比流及多项式自排斥流。
We study a simple stochastic differential equation driven by one Brownian motion on a general oriented metric graph whose solutions are stochastic flows of kernels. Under some condition, we describe the laws of all solutions. This work is a natural continuation of some previous papers by Hajri, Hajri-Raimond and Le Jan-Raimond where some particular graphs have been considered.
The weak limits of the measure-valued processes organized as a mass carried by the interacting Brownian particles are described. As a limiting flow the Arrattia flow is obtained.
We show that for a large class of stochastic flows the spatial derivative grows at most exponentially fast even if one takes the supremum over a bounded set of initial points. We derive explicit bounds on the growth rates that depend on the local characteristics of the flow and the box dimension of the set.
A stochastic flow of homeomorphisms of the real line previously studied by Bass and Burdzy is shown to arise in describing a Brownian motion conditional on knowing its local times on hitting a fixed level. This makes it possible to connect Ray-Knight type results for the flow with the classical Ray-Knight theorems for Brownian motion.
In this article we study transformations of Gaussian field by stochastic flow on the plane. A stochastic flow is a solution to the equation with interaction whose coefficients depend on the occupation measure of the field. We consider nonsmooth Gaussian field, which has self-intersection local times of any multiplicity. In the article we prove the existence of self-intersection local times for the transformed field and study its asymptotics.
Given a one-dimensional stochastic differential equation, one can associate to this equation a stochastic flow on $[0,+\infty )$, which has an absorbing barrier at zero. Then one can define its dual stochastic flow. In \cite{AW}, Akahori and Watanabe showed that its one-point motion solves a corresponding stochastic differential equation of Skorokhod-type. In this paper, we consider a discrete-time stochastic-flow which approximates the original stochastic flow. We show that under some assumptions, one-point motions of its dual flow also approximates the corresponding reflecting diffusion. We …
The Large Deviation Principle for Stochastic Flow of Stochastic Slow-Fast Motions
本文研究完全耦合慢快运动中慢变量的随机流，证明其存在性并满足大偏差原理，方法基于Khasminskii平均原理、变分表示和弱收敛框架。
Stochastic flows generated by reflected SDEs in a half-plane with an additive diffusion term are considered. A derivative in the initial data is represented a.s. as an infinite product of matrices. We use this representation and construct an example of a reflecting flow with a linear drift such that it is not locally continuously differentiable.
Fast advection asymptotics for a stochastic reaction-diffusion-advection equation are studied in this paper. To describe the asymptotics, one should consider a suitable class of SPDEs defined on a graph, corresponding to the stream function of the underlying incompressible flow.
Gaussian structure in coalescing stochastic flows
本文针对对应Arratia流的点测度开展研究，推导出关于该测度的多重积分的中心极限定理，为相关随机流领域的概率分析提供了理论支撑。
Stochastic heat flow is a black noise
本文证明随机热流是黑噪声，并推论由平滑时空白噪声驱动的2d随机热方程在标度极限下渐近独立于该噪声。
In this paper we compute the level-crossing intensity for the density of the image of the Lebesgue measure under the action of a Brownian stochastic flow which is a smooth approximation of the Arratia flow and determine its asymptotic behaviour as the height of the level tends to infinity.
We prove that a stochastic flow of reflected Brownian motions in a smooth multidimensional domain is differentiable with respect to its initial position. The derivative is a linear map represented by a multiplicative functional for reflected Brownian motion. The method of proof is based on excursion theory and analysis of the deterministic Skorokhod equation.
We consider the Itô SDE with partially Sobolev coefficients. Under some suitable conditions, we show the existence, uniqueness and stability of generalized stochastic flows associated to such an equation. As an application, we prove the weak differentiability of the stochastic flow generated by the Itô SDE with Sobolev coefficients.
Some topological properties of stochastic flow $φ_t(x)$ generated by stochastic differential equation in a ${\mathbb R}^d_+$ with normal reflection at the boundary are investigated. Sobolev differentiability in initial condition is received. The absolute continuity of the measure-valued process $μ\circφ_t^{-1}$, where $μ\llλ^d,$ is studied.
We introduce and characterize a class of flows, which turn out to be Gaussian. This characterization allows us to show, using the Monotonicity inequality, that the transpose of the flow, for an extended class of initial conditions, is the unique solution of the SPDE introduced in Rajeev and Thangavelu (2008).
We study the existence of the stochastic flow associated to a linear stochastic evolution equation $$d X= AX\,d t +\sum_{k} B_k X\,d W_k, $$ on a Hilbert space. Our first result covers the case where $A$ is the generator of a $C_0$-semigroup, and $(B_k)$ is a sequence of bounded linear operators such that $\sum_k\|B_k\|<+\infty$. We also provide sufficient conditions for the existence of stochastic flows in the Schatten classes beyond the space of Hilbert-Schmidt operators. Some new results and examples concerning the so-called commutative case are presented as well.
Complexity of flows with interactions and stochastic analysis
本文通过核提升将测度值随机方程表示为Hilbert值马尔可夫过程，确定生成元并建立稠密值域条件，为分析相互作用随机方程的稳定性提供框架。
In this paper we prove the stochastic homeomorphism flow property and the strong Feller property for stochastic differential equations with sigular time dependent drifts and Sobolev diffusion coefficients. Moreover, the local well posedness under local assumptions are also obtained. In particular, we extend Krylov and Röckner's results in \cite{Kr-Ro} to the case of non-constant diffusion coefficients.
We study diffusion processes and stochastic flows which are time-changed random perturbations of a deterministic flow on a manifold. Using non-symmetric Dirichlet forms and their convergence in a sense close to the Mosco-convergence, we prove that, as the deterministic flow is accelerated, the diffusion process converges in law to a diffusion defined on a different space. This averaging principle also holds at the level of the flows. Our contributions in this article include: a proof of an original averaging principle for stochastic flows of kernels; the definition and study of a convergence o…
Motivated from time-inconsistent stochastic control problems, we introduce a new type of coupled forward-backward stochastic systems, namely, flows of forward-backward stochastic differential equations. They are systems consisting of a single forward SDE and a continuum of BSDEs, which are defined on different time-intervals and connected via an equilibrium condition. We formulate a notion of equilibrium solutions in a general framework and prove small-time well-posedness of the equations. We also consider discretized flows and show that their equilibrium solutions approximate the original one…
Stochastic Curve Shortening Flow with Scale-Dependent Noise
本文研究了由尺度依赖噪声扰动的随机曲线缩短流，通过将问题转化为拟线性演算方程，获得了局部强解的存在性和唯一性。
In this paper we prove, for small Hurst parameters, the higher order differentiability of a stochastic flow associated with a stochastic differential equation driven by an additive multi-dimensional fractional Brownian noise, where the bounded variation part is given by the local time of the unknown solution process. The proof of this result relies on Fourier analysis based variational calculus techniques and on intrinsic properties of the fractional Brownian motion.
In this paper we have constructed an approximation for the Harris flow and the Arratia flow using a sequence of independent stationary Gaussian processes as a perturbation. We have established what should be the relationship between the step of approximation and smoothness of the covariance of the perturbing processes in order to have convergence of the approximating functions to the Arratia flow.
We establish moment estimates for the invariant measure of a stochastic partial differential equation describing motion by mean curvature flow in (1+1) dimension, leading to polynomial stability of the associated Markov semigroup. We also prove maximal dissipativity for the related Kolmogorov operator.
In this paper we construct an object which we call the full Brownian web (FBW) and prove that the collection of all space-time trajectories of a class of one-dimensional stochastic flows converges weakly, under diffusive rescaling, to the FBW. The (forward) paths of the FBW include the coalescing Brownian motions of the ordinary Brownian web along with bifurcating paths. Convergence of rescaled stochastic flows to the FBW follows from general characterization and convergence theorems that we present here combined with earlier results of Piterbarg.
Robust estimates for the performance of complicated queueing networks can be obtained by showing that the number of jobs in the network is stochastically comparable to a simpler, analytically tractable reference network. Classical coupling results on stochastic ordering of network populations require strong monotonicity assumptions which are often violated in practice. However, in most real-world applications we care more about what goes through a network than what sits inside it. This paper describes a new approach for ordering flows instead of populations by augmenting network states with th…
The probabilistic equivalent formulation of Dupire's PDE is the Put-Call duality equality. In local volatility models including exponential Lévy jumps, we give a direct probabilistic proof for this result based on stochastic flows arguments. This approach also enables us to check the probabilistic equivalent formulation of various generalizations of Dupire's PDE recently obtained by Pironneau by the adjoint equation technique in the case of complex options.
We show the existence of superprocesses in a random medium with location dependent branching. Technically, we make use of a duality relation to establish the uniqueness of the martingale problem and to obtain the moment formulas.
This is a brief survey of T. E. Harris's work on recurrent Markov processes and on stochastic flows, and of some more recent work in these fields.
Fractional Moments of Small-Ball Masses for the Stochastic Heat Flow
针对临界二维随机热流的收缩球质量，研究人员基于Gu和Tsai的几何块分解，引入分数乘积测度变换，证明了其对数分数矩指数。
We derive sufficient conditions for the differentiability of all orders for the flow of stochastic differential equations with jumps, and prove related $L^p$-integrability results for all orders. Our results extend similar results obtained in [Kun04] for first order differentiability and rely on the Burkholder-Davis-Gundy inequality for time inhomogeneous Poisson random measures on ${\Bbb R}_+\times {\Bbb R}$, for which we provide a new proof.
In this paper, we first explore certain structural properties of Lévy flows and use this information to obtain the existence of strong solutions to a class of Stochastic PDEs in the space of tempered distributions, driven by Lévy noise. The uniqueness of the solutions follows from Monotonicity inequality. These results extend an earlier work Bhar (2017) on the diffusion case.
Stochastic conformal flows in even dimensions
研究偶数维流形上Q-曲率流的两种随机模拟，利用狄利克雷形式理论构造弱解，并证明Liouville Q流是偶数维Polyakov-Liouville测度的随机量子化。
A universality result for the critical 2d stochastic heat flow
本文证明一大类满足线性流性质的离散与连续模型在扩散标度下收敛到临界二维随机热流，建立了普适性结果。
Splitting for some classes of homeomorphic and coalescing stochastic flows
针对Th.E. Harris型带公共噪声的随机流，将Kato-Trotter分裂格式应用于同胚与聚合随机流，证明有限维运动弱收敛性，给出推前测度与对偶流收敛结果，并借助与Euler-Maruyama格式的相似性在正则假设下估计收敛速度。
We define an equation on a simple graph which is an extension of Tanaka equation and the skew Brownian motion equation. We then apply the theory of transition kernels developped by Le Jan and Raimond and show that all the solutions can be classified by probability measures.
We consider a $d$-dimensional SDE with an identity diffusion matrix and a drift vector being a vector function of bounded variation. We give a representation for the derivative of the solution with respect to the initial data.
Following Le Jan and Watanabe we define a connection associated with a non-degenrrate diffusion operators. This connection is characterized here and shown to be the Levi-Civita connection for gradient systems. This both explains why such systems have useful properties and allows us to extend these properties to more general systems. Topics described here include: moment estimates for $Tξ_t$, a Weitzenböck formula for the generator of the semigroup on p-forms induced by the flow, a Bismut type formula for $d\log p_t$ in terms of an arbitrary metric connection, and a generalized Bochner vanishin…
This paper examines the existence of the self-intersection local time for a superprocess over a stochastic flow in dimensions $d\leq3$, which through constructive methods, results in a Tanaka-like representation. The superprocess over a stochastic flow is a superprocess with dependent spatial motion, and thus Dynkin's proof of existence, which requires multiplicity of the log-Laplace functional, no longer applies. Skoulakis and Adler's method of calculating moments is extended to higher moments, from which existence follows.
Using Huisken results about the mean curvature flow on a strictly convex hypersurface, and Kendall-Cranston coupling, we will build a stochastic process without birth, and show that there exists a unique law of such process. This process has many similarities with the circular Brownian motions studied by Émery, Schachermayer, and Arnaudon. In general, this process is not a stationary process, it is linked with some differential equation without initial condition. We will show that this differential equation has a unique solution up to a multiplicative constant.
Maximum principle for optimal control of interacting particle system: stochastic flow model
针对相互作用粒子系统的随机最优控制问题，本文引入带相互作用的广义倒向随机微分方程，建立随机最大值原理及解的存在唯一性，并通过线性二次情形验证结果。
Existence of random dynamical systems for a class of coalescing stochastic flows on $\mathbb{R}$ is proved. A new state space for coalescing flows is built. As particular cases coalescing flows of solutions to stochastic differential equations independent before meeting time and coalescing Harris flows are considered.
For a class of coalescing stochastic flows on the real line the existence of dual flows is proved. A stochastic flow and its dual are constructed as a forward and backward perfect cocycles over the same metric dynamical system. The metric dynamical system itself is defined on a new state space for coalescing flows. General results are applied to Arratia flows with drift.
In this article I will prove new representation for the Levi-Civita connection in terms of the stochastic flow corresponding to Brownian motion on manifold.
The paper generalizes the construction by stochastic flows of consistent utility processes introduced by M. Mrad and N. El Karoui in (2010). The utilities random fields are defined from a general class of processes denoted by $\GX$. Making minimal assumptions and convex constraints on test-processes, we construct by composing two stochastic flows of homeomorphisms, all the consistent stochastic utilities whose the optimal-benchmark process is given, strictly increasing in its initial condition. Proofs are essentially based on stochastic change of variables techniques.
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