arXiv subjects

Zhongmin Qian

Publications and source records attributed to Zhongmin Qian.

60 records · Page 4Linked to original sources

arXiv2011

Differential Structure and Flow equations on Rough Path Space

We introduce a differential structure for the space of weakly geometric p rough paths over a Banach space V for 2<p<3. We begin by considering a certain natural family of smooth rough paths and differentiating in the truncated tensor series. The resulting object has a clear interpretation, even for non-smooth rough paths, which we take to be an element of the tangent space. We can associate it uniquely to an equivalence class of curves, with equivalence defined by our differential structure. Thus, for a functional on rough path space, we can define the derivative in a tangent direction analogous to defining the derivative in a Cameron-Martin direction of a functional on Wiener space. Our tangent space contains many more directions than the Cameron-Martin space and we do not require quasi-invariance of Wiener measure. In addition we also locally (globally) solve the associated flow equation for a class of vector fields satisfying a local (global) Lipshitz type condition.

math.PR↗

arXiv2011

Stratonovich's Signatures of Brownian Motion Determine Brownian Sample Paths

The signature of Brownian motion in $\mathbb{R}^{d}$ over a running time interval $[0,T]$ is the collection of all iterated Stratonovich path integrals along the Brownian motion. We show that, in dimension $d\geq 2$, almost all Brownian motion sample paths (running up to time $T$) are determined by its signature over $[0,T]$

math.PR↗

arXiv2009

Non-Linear Evolution Equations Driven by Rough Paths

We prove existence and uniqueness results for (mild) solutions to some non-linear parabolic evolution equations with a rough forcing term. Our method of proof relies on a careful exploitation of the interplay between the spatial and time regularity of the solution by capitialising some of Kato's ideas in semigroup theory. Classical Young integration theory is then shown to provide a means of interpreting the equation. As an application we consider the three dimensional Navier-Stokes system with a stochastic forcing term arising from a fractional Brownian motion with h > 1/2.

math.PR↗

arXiv2009

Martingale representations for diffusion processes and backward stochastic differential equations

In this paper we explain that the natural filtration of a continuous Hunt process is continuous, and show that martingales over such a filtration are continuous. We further establish a martingale representation theorem for a class of continuous Hunt processes under certain technical conditions. In particular we establish the martingale representation theorem for the martingale parts of (reflecting) symmetric diffusions in a bounded domain with a continuous boundary. Together with an approach put forward in Lyons et al(2009), our martingale representation theorem is then applied to the study of initial and boundary problems for quasi-linear parabolic equations by using solutions to backward stochastic differential equations over the filtered probability space determined by reflecting diffusions in a bounded domain with only continuous boundary.

math.PR↗

arXiv2008

The Navier-Stokes Equations with the Kinematic and Vorticity Boundary Conditions on Non-Flat Boundaries

We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in $\R^n$ with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure $p$ can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition (without the incompressibility condition), which establishes a velocity mapping. Then we develop \emph{apriori} estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in $\R^n (n\ge 3)$ with nonhomogeneous vorticity boundary condition converges in $L^2$ to the corresponding Euler equations satisfying the kinematic condition.

math.AP↗

arXiv2008

A Study of the Navier-Stokes Equations with the Kinematic and Navier Boundary Conditions

We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a domain in $\R^3$ with compact and smooth boundary, subject to the kinematic and Navier boundary conditions. We first reformulate the Navier boundary condition in terms of the vorticity, which is motivated by the Hodge theory on manifolds with boundary from the viewpoint of differential geometry, and establish basic elliptic estimates for vector fields subject to the kinematic and Navier boundary conditions. Then we develop a spectral theory of the Stokes operator acting on divergence-free vector fields on a domain with the kinematic and Navier boundary conditions. Finally, we employ the spectral theory and the necessary estimates to construct the Galerkin approximate solutions and establish their convergence to global weak solutions, as well as local strong solutions, of the initial-boundary problem. Furthermore, we show as a corollary that, when the slip length tends to zero, the weak solutions constructed converge to a solution to the incompressible Navier-Stokes equations subject to the no-slip boundary condition for almost all time. The inviscid limit of the strong solutions to the unique solutions of the initial-boundary value problem with the slip boundary condition for the Euler equations is also established.

math.AP↗