Feng-Yu Wang
By using Hsu's multiplicative functional for the Neumann heat equation, a natural damped gradient operator is defined for the reflecting Brownian motion on compact manifolds with boundary. This operator is linked to quasi-invariant flows in terms of a integration by parts formula, which leads to the standard log-Sobolev inequality for the associated Dirichlet form on the path space.
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For sub-Riemannian manifolds with a chosen complement, we first establish the derivative formula and integration by parts formula on path space with respect to a natural gradient operator. By using these formulae, we then show that upper and lower bounds of the horizontal Ricci curvature correspond to functional inequalities on path space analogous to what has been established in Riemannian geometry by Aaron Naber, such as gradient inequalities, log-Sobolev and Poincaré inequalities.
Let $L=\DD+Z$ for a $C^1$ vector field $Z$ on a complete Riemannian manifold possibly with a boundary. By using the uniform distance, a number of transportation-cost inequalities on the path space for the (reflecting) $L$-diffusion process are proved to be equivalent to the curvature condition $\Ric-\nn Z\ge - K$ and the convexity of the boundary (if exists). These inequalities are new even for manifolds without boundary, and are partly extended to non-convex manifolds by using a conformal change of metric which makes the boundary from non-convex to convex.
Higher-order variation and pathwise Ito calculus on manifolds
该研究针对流形上任意低正则性路径发展内禀微积分，将Föllmer逐段伊藤微积分扩展到流形值路径，适用于不规则随机过程，以分数布朗运动提升为例说明其应用。
Certain natural geometric approximation schemes are developed for Wiener measure on a compact Riemannian manifold. These approximations closely mimic the informal path integral formulas used in the physics literature for representing the heat semi-group on Riemannian manifolds. The path space is approximated by finite dimensional manifolds consisting of piecewise geodesic paths adapted to partitions $P$ of $[0,1]$. The finite dimensional manifolds of piecewise geodesics carry both an $H^{1}$ and a $L^{2}$ type Riemannian structures $G^i_P$. It is proved that as the mesh of the partition tends …
In this paper, we prove a finite dimensional approximation scheme for the Wiener measure on closed Riemannian manifolds, establishing a generalization for $L^{1}$-functionals, of the approach followed by Andersson and Driver on [1]. We follow a new approach motived by categorical concept of colimit.
In this article, we continue the discussion of Fang-Wu (2015) to estimate the spectral gap of the Ornstein-Uhlenbeck operator on path space over a Riemannian manifold of pinched Ricci curvature. Along with explicit estimates we study the short-time asymptotics of the spectral gap. The results are then extended to the path space of Riemannian manifolds evolving under a geometric flow. Our paper is strongly motivated by Naber's recent work (2015) on characterizing bounded Ricci curvature through stochastic analysis on path space.
Fang-Wu\cite{FW17} presented a explicit spectral gap for the O-U process on path space over a Riemannian manifold without boundary under the bounded Ricci curvature conditions. In this paper, we will extend these results to the case of the Riemannian manifold with boundary. Moreover, we also derive the similar results for the stochastic heat process.
Some parts of stochastic analysis on curved spaces are revisted. A concise proof of the quasi-invariance of the Wiener measure on the path spaces over a Riemannian manifold is presented. The shifts are allowed to be in the Cameron-Martin space and random. The second part of the paper presents some remarks on the anticipative integrals on Riemannian manifolds.
In prior work \cite{AD} of Lars Andersson and Bruce K. Driver, the path space with finite interval over a compact Riemannian manifold is approximated by finite dimensional manifolds $H_{x,¶} (M)$ consisting of piecewise geodesic paths adapted to partitions $¶$ of $[0,T]$, and the associated Wiener measure is also approximated by a sequence of probability measures on finite dimensional manifolds. In this article, we will extend their results to the general path space(possibly with infinite interval) over a non-compact Riemannian manifold by using the cutoff method of compact Riemannian manifold…
We prove the existence of the O-U Dirichlet form and the damped O-U Dirichlet form on path space over a general non-compact Riemannian manifold which is complete and stochastically complete. We show a weighted log-Sobolev inequality for the O-U Dirichlet form and the (standard) log-Sobolev inequality for the damped O-U Dirichlet form. In particular, the Poincaré inequality (and the super Poincaré inequality) can be established for the O-U Dirichlet form on path space over a class of Riemannian manifolds with unbounded Ricci curvatures. Moreover, we construct a large class of quasi-regular loca…
Brownian motion on manifolds with non-trivial diffusion coefficient can be constructed by stochastic development of Euclidean Brownian motions using the fiber bundle of linear frames. We provide a comprehensive study of paths for such processes that are most probable in the sense of Onsager-Machlup, however with path probability measured on the driving Euclidean processes. We obtain both a full characterization of the resulting family of most probable paths, reduced equation systems for the path dynamics where the effect of curvature is directly identifiable, and explicit equations in special …
Let M be a compact Riemannian manifold without boundary and let H be a self-adjoint generalized Laplace operator acting on sections in a bundle over M. We give a path integral formula for the solution to the corresponding heat equation. This is based on approximating path space by finite dimensional spaces of geodesic polygons. We also show a uniform convergence result for the heat kernels. This yields a simple and natural proof for the Hess-Schrader-Uhlenbrock estimate and a path integral formula for the trace of the heat operator.
Cutoff for geodesic paths on hyperbolic manifolds
本文证明任意紧双曲流形上从局部化初始条件出发的测地路径具有截止现象，并将Golubev和Kamber关于大体积双曲曲面的结果推广到任意维数，方法基于Lubetzky-Peres谱策略和球面平均算子的谱分析。
In this paper, we build the foundation for a theory of controlled rough paths on manifolds. A number of natural candidates for the definition of manifold valued controlled rough paths are developed and shown to be equivalent. The theory of controlled rough one-forms along such a controlled path and their resulting integrals are then defined. This general integration theory does require the introduction of an additional geometric structure on the manifold which we refer to as a "parallelism." The transformation properties of the theory under change of parallelisms is explored. Using these trans…
Let $L_t:=Δ_t+Z_t$ for a $C^{1,1}$-vector field $Z$ on a differential manifold $M$ with possible boundary $\partial M$, where $Δ_t$ is the Laplacian induced by a time dependent metric $g_t$ differentiable in $t\in [0,T_c)$. We first introduce the damp gradient operator, defined on the path space with reference measure $\mathbb{P}$, the law of the (reflecting) diffusion process generated by $L_t$ on the base manifold; then establish the integration by parts formula for underlying directional derivatives and prove the log-Sobolev inequality for the associated Dirichlet form, which is further app…
Since the breakthrough in rough paths theory for stochastic ordinary differential equations (SDEs), there has been a strong interest in investigating the rough differential equation (RDE) approach and its numerous applications. Rough path techniques can stay closer to deterministic analytical methods and have the potential to transfer many pathwise ordinary differential equation (ODE) techniques more directly to a stochastic setting. However, there are few works that analyze dynamical properties of RDEs and connect the rough path / regularity structures, ODE and random dynamical systems approa…
On the use of the Belopol'skaya-Daletskii representation of a diffusion on a Riemann manifold to construct path integrals
研究利用黎曼流形上随机微分方程的别洛波尔斯卡娅 - 达列茨基公式，通过指数映射描述扩散增量，构建扩散路径测度有限维近似的等变表示。
Rough differential equations on manifolds via natural bundles
本文利用自然丛理论，开发了由全分支粗糙路径驱动的流形上的粗糙微分方程新框架，证明了解的存在唯一性，给出子流形不变的充要条件，无需额外结构。
We introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural. On the way we develop a notion of rough integration and an efficient and intrinsic theory of rough differential equations (RDEs) on manifolds. The theory of RDEs is then used to construct parallel translation along manifold valued rough paths. Finally, this framework is used to show there is a one to one correspondence between rough paths on a d-dimensional manifold and rough paths on d-dimensional Euclidean space. This last result is a rough path analogue of Cartan's develo…
Taylor-like approximations of center manifolds for rough differential equations
针对几何粗糙路径驱动的粗糙微分方程，结合粗糙路径与确定性中心流形理论推导其局部随机中心流形的类泰勒逼近，逼近系数为同驱动RDEs的平稳解，并通过带乘性噪声的SDEs验证方法。
A typical path integral on a manifold, $M$ is an informal expression of the form \frac{1}{Z}\int_{σ\in H(M)} f(σ) e^{-E(σ)}\mathcal{D}σ, \nonumber where $H(M)$ is a Hilbert manifold of paths with energy $E(σ) < \infty$, $f$ is a real valued function on $H(M)$, $\mathcal{D}σ$ is a \textquotedblleft Lebesgue measure \textquotedblright and $Z$ is a normalization constant. For a compact Riemannian manifold $M$, we wish to interpret $\mathcal{D}σ$ as a Riemannian \textquotedblleft volume form \textquotedblright over $H(M)$, equipped with its natural $G^{1}$ metric. Given an equally spaced partition…
Most probable paths for developed processes
本文提出了一种通用方法，通过测量Onsager-Machlup函数在反发展上的值，定义和识别最可能路径，适用于多种流形值过程，如Lie群、同质空间和形状分析中的地标空间。
In this paper, we prove the existence of martingale solutions to the stochastic heat equation taking values in a Riemannian manifold, which admits Wiener (Brownian bridge) measure on the Riemannian path (loop) space as an invariant measure using a suitable Dirichlet form. Using the Andersson-Driver approximation, we heuristically derive a form of the equation solved by the process given by the Dirichlet form. Moreover, we establish the log-Sobolev inequality for the Dirichlet form in the path space. In addition, some characterizations for the lower bounds of the Ricci curvature are presented r…
We define horizontal diffusion in $C^1$ path space over a Riemannian manifold and prove its existence. If the metric on the manifold is developing under the forward Ricci flow, horizontal diffusion along Brownian motion turns out to be length preserving. As application, we prove contraction properties in the Monge-Kantorovich minimization problem for probability measures evolving along the heat flow. For constant rank diffusions, differentiating a family of coupled diffusions gives a derivative process with a covariant derivative of finite variation. This construction provides an alternative m…
Malliavin Calculus can be seen as a differential calculus on Wiener spaces. We present the notion of stochastic manifold for which the Malliavin Calculus plays the same role as the classical differential calculus for the differential manifolds. The set of the paths in a Riemmanian compact manifold is then seen as a particular case of the above structure.
This paper gives a rigorous interpretation of a Feynman path integral on a Riemannian manifold M with non-positive sectional curvature. A $L^2$ Riemannian metric $G_P$ is given on the space of piecewise geodesic paths $H_P(M)$ adapted to the partition $P$ of $[0,1]$, whence a finite-dimensional approximation of Wiener measure is developed. It is proved that, as $mesh(P) \to 0$, the approximate Wiener measure converges in a $L^1$ sense to the measure $e^{-\frac{2 + \sqrt{3}}{20\sqrt{3}} \int_0^1 Scal(σ(s)) ds} dν(σ)$ on the Wiener space $W(M)$ with Wiener measure $ν$. This gives a possible pres…
An L2 theory of differential forms is proposed for the Banach manifold of continuous paths on Riemannian manifolds M furnished with its Brownian motion measure. Differentiation must be restricted to certain Hilbert space directions, the H-tangent vectors. To obtain a closed exterior differential operator the relevant spaces of differential forms, the H-forms, are perturbed by the curvature of M. A Hodge decomposition is given for L2 H-one-forms, and the structure of H-two -forms is described. The dual operator d* is analysed in terms of a natural connection on the H-tangent spaces. Malliavin c…
Extensions of the fractional Brownian fields are constructed over a complete Riemannian manifold. This construction is carried out for the full range of the Hurst parameter $α\in(0,1)$. In particular, we establish existence, distributional scaling (self-similiarity), stationarity of the increments, and almost sure Hölder continuity of sample paths. Stationary counterparts to these fields are also constructed.
We develop a path integral framework for determining most probable paths in a class of systems of stochastic differential equations with piecewise-smooth drift and additive noise. This approach extends the Freidlin-Wentzell theory of large deviations to cases where the system is piecewise-smooth and may be non-autonomous. In particular, we consider an $n-$dimensional system with a switching manifold in the drift that forms an $(n-1)-$dimensional hyperplane and investigate noise-induced transitions between metastable states on either side of the switching manifold. To do this, we mollify the dr…
We prove a center manifold theorem for rough partial differential equations (rough PDEs). The class of rough PDEs we consider contains as a key subclass reaction-diffusion equations driven by nonlinear multiplicative noise, where the stochastic forcing is given by a $γ$-Hölder rough path, for $γ\in(1/3,1/2]$. Our proof technique relies upon the theory of rough paths and analytic semigroups in combination with a discretized Lyapunov-Perron-type method in a suitable scale of interpolation spaces. The resulting center manifold is a random manifold in the sense of the theory of random dynamical sy…
The wave maps equation and Brownian paths
本文研究取值于紧黎曼流形的$(1+1)$维波映射方程，以布朗路径为初始数据，证明了相关初值问题的概率局部适定性，结合了解析、几何与概率方法。
By the Lyapunov-Perron method,we prove the existence of random inertial manifolds for a class of equations driven simultaneously by non-autonomous deterministic and stochastic forcing. These invariant manifolds contain tempered pullback random attractors if such attractors exist. We also prove pathwise periodicity and almost periodicity of inertial manifolds when non-autonomous deterministic forcing is periodic and almost periodic in time, respectively.
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