含路径依赖有界变差项的粗糙微分方程

Global existence for rough differential equations under linear growth conditions

We prove existence of global solutions for differential equations driven by a geometric rough path under the condition that the vector fields have linear growth. We show by an explicit counter-example that the linear growth condition is not sufficient if the driving rough path is not geometric. This settle a long-standing open question in the theory of rough paths. So in the geometric setting we recover the usual sufficient condition for differential equation. The proof rely on a simple mapping of the differential equation from the Euclidean space to a manifold to obtain a rough differential e…

由噪声驱动的Gaussian粗糙路径驱动的粗糙微分方程的正则化

Regularization by noise for rough differential equations driven by Gaussian rough paths

本文研究了由Gaussian粗糙路径驱动的粗糙微分方程，在非确定性和一致椭圆性条件下证明了低正则性漂移的逐路径适定性，并利用流变换和Malliavin计算确定了分数布朗运动下漂移的Hölder连续性界限。

Quantitative John--Nirenberg inequality for stochastic processes of bounded mean oscillation

Stroock and Varadhan in 1997 and Geiss in 2005 independently introduced stochastic processes with bounded mean oscillation (BMO) and established their exponential integrability with some unspecified exponential constant. This result is an analogue of the John--Nirenberg inequality for functions of bounded mean oscillation. In this work, we quantify the size of the exponential constant by the modulus of mean oscillation. Some new applications of BMO processes in rough stochastic differential equations, numerical approximations and regularization by noise are discussed.

Boundary Behavior of Harmonic Functions for Truncated Stable Processes

For any αin (0, 2), a truncated symmetric α-stable process is a symmetric Levy process with no diffusion part and with a Levy density given by c|x|^{-d-α} 1_{|x|< 1} for some constant c. In previous paper we have studied the potential theory of truncated symmetric stable processes. Among other things, we proved that the boundary Harnack principle is valid for the positive harmonic functions of a truncated symmetric stable process in any bounded convex domain and showed that the Martin boundary of any bounded convex domain with respect to this process is the same as the Euclidean boundary. Howe…

Geometric rough paths on infinite dimensional spaces

Similar to ordinary differential equations, rough paths and rough differential equations can be formulated in a Banach space setting. For $α\in (1/3,1/2)$, we give criteria for when we can approximate Banach space-valued weakly geometric $α$-rough paths by signatures of curves of bounded variation, given some tuning of the Hölder parameter. We show that these criteria are satisfied for weakly geometric rough paths on Hilbert spaces. As an application, we obtain Wong-Zakai type result for function space valued martingales using the notion of (unbounded) rough drivers.

Rough Hypoellipticity for the Heat Equation in Dirichlet Spaces

This paper aims at proving the local boundedness and continuity of solutions of the heat equation in the context of Dirichlet spaces under some rather weak additional assumptions. We consider symmetric local regular Dirichlet forms which satisfy mild assumptions concerning (a) the existence of cut-off functions, (b) a local ultracontractivity hypothesis, and (c) a weak off-diagonal upper bound. In this setting, local weak solutions of the heat equation, and their time derivatives, are shown to be locally bounded; they are further locally continuous, if the semigroup admits a locally continuous…

A Note on the Notion of Geometric Rough Paths

We use simple sub-Riemannian techniques to prove that an arbitrary geometric p-rough path in the sense of Lyons (98) is the limit in sup-norm of a sequence of canonically lifted smooth paths, which are uniformly bounded in p-variation, clarifying the two different definitions of a geometric p-rough path, Lyons (98), Lyons/Qian (02). Our proofs are based on fine estimates in terms of control functions and are sufficiently general to include the case of Hoelder- and modulus-type regularity, Friz/Victoir (03). This allows us to extend a few classical results on Hoelder-spaces (Ciesielski, Musiela…

变分性与具有不规则系数的粗糙积分的存在性

Variability and the existence of rough integrals with irregular coefficients

本文在分数阶微积分的粗糙路径分析框架下，利用变分性证明了在Lipschitz系数一阶偏导数为有界变差时，关于Hölder连续乘性泛函积分的存在性，并应用于Hurst指数在1/3到1/2之间的fractional Brownian motions等Gaussian过程。

Speed of propagation for Hamilton-Jacobi equations with multiplicative rough time dependence and convex Hamiltonians

We show that the initial value problem for Hamilton-Jacobi equations with multiplicative rough time dependence, typically stochastic, and convex Hamiltonians satisfies finite speed of propagation. We prove that in general the range of dependence is bounded by a multiple of the length of the "skeleton" of the path, that is a piecewise linear path obtained by connecting the successive extrema of the original one. When the driving path is a Brownian motion, we prove that its skeleton has almost surely finite length. We also discuss the optimality of the estimate.

Non-uniqueness for reflected rough differential equations

We give an example of a reflected diffferential equation which may have infinitely many solutions if the driving signal is rough enough (e.g. of infinite $p$-variation, for some $p>2$). For this equation, we identify a sharp condition on the modulus of continuity of the signal under which uniqueness holds. Lévy's modulus for Brownian motion turns out to be a boundary case. We further show that in our example, non-uniqueness holds almost surely when the driving signal is a fractional Brownian motion with Hurst index $H < \frac{1}{2}$. The considered equation is driven by a two-dimensional signa…

测量信号的粗糙度

Measuring the roughness of a signal

本文提出基于归一化幂变差的非参数方法估计路径粗糙度，在Hölder正则等条件下证明一致性并给出收敛速度，适用于分数布朗运动等，并通过数值实验验证。

签名核与施温-戴森核方程作为双参数粗糙微分方程

Signature Kernel and Schwinger-Dyson Kernel Equations as Two-Parameter Rough Differential Equations

本文提出双参数粗糙微分方程框架，研究签名核与施温-戴森核方程的性质，建立鲁棒的双参数粗糙积分理论，并推导数值方案及复杂度分析。