Zhongmin Qian

System control and rough paths

0. Preface 1. Introduction 2. Lipschitz paths 3. Rough paths 4. Brownian rough paths 5. Path integration along rough paths 6. Universal limit theoem 7. Vector fields and Flow Equations

Large deviations and support theorem for diffusion processes via rough paths

We use the continuity theorem of Lyons for rough paths in the p-variation topology to produce an elementary approach to the large deviation principle and the support theorem for diffusion processes. The proofs reduce to establish the corresponding results for Brownian motion itself as a rough path in the p-variation topology, 2

Harnack inequalities on a manifold with positive or negative Ricci curvature

Several new Harnack estimates for positive solutions of the heat equation on a complete Riemannian manifold with Ricci curvature bounded below by a positive (or a negative) constant are established. These estimates are sharp both for small time, for large time and for large distance, and lead to new estimates for the heat kernel of a manifold with Ricci curvature bounded below

Stochastic differential equations for fractional Brownian motions

Abstract We construct via dyadic approximations a canonical geometric rough path in the sense of [7] associated to a fractional Brownian motion with Hurst parameter h, h∈ ]1/4,1/2[ . Therefore, we obtain a Wong–Zakai type approximation theorem for solutions of stochastic differential equations driven by these fractional Brownian motions.

A Comparison Theorem for an Elliptic Operator

In this note, using Γ2 (Bakry–Emery curvature operator) and the classical maximum principle, we establish a comparison estimate for a general elliptic operator on a manifold, without using Jacobi field theory.

Existence of Solutions to a Class of Indefinite Stochastic Riccati Equations

An indefinite stochastic Riccati equation is a matrix-valued, highly nonlinear backward stochastic differential equation together with an algebraic, matrix positive definiteness constraint. We introduce a new approach to solve a class of such equations (including the existence of solutions) driven by one-dimensional Brownian motion. The idea is to replace the original equation by a system of backward stochastic differential equations (without involving any algebraic constraint) whose existence of solutions automatically enforces the original algebraic constraint to be satisfied.

The Navier-Stokes equations with the kinematic and vorticity boundary conditions on non-flat boundaries

Abstract We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in ℝn 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 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 ℝn(n ≥ 3) with nonhomogeneous vorticity boundary condition converge in L2 to the corresponding Euler equations satisfying the kinematic condition.

On Harnack estimates for positive solutions of the heat equation on a complete manifold

Abstract We establish several new Harnack estimates for the nonnegative solutions of the heat equation on a complete Riemannian manifold with Ricci curvature bounded by a positive or negative constant. This extends to symmetric diffusions whose generator satisfies a “curvature-dimension” inequality.

On an inversion theorem for Stratonovich’s signatures of multidimensional diffusion paths

In the present paper, we prove that with probability one, the Stratonovich signatures of a multidimensional diffusion process (possibly degenerate) over [0,1], which is the collection of all iterated Stratonovichs integrals of the diffusion process over [0,1], determine the diffusion sample paths.

G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion

The present article is devoted to the study of sample paths of G-Brownian motion and stochastic differential equations (SDEs) driven by G-Brownian motion from the viewpoint of rough path theory. As the starting point, by using techniques in rough path theory, we show that quasi-surely, sample paths of G-Brownian motion can be enhanced to the second level in a canonical way so that they become geometric rough paths of roughness 2 < p < 3. This result enables us to introduce the notion of rough differential equations (RDEs) driven by G-Brownian motion in the pathwise sense under the general framework of rough paths. Next we establish the fundamental relation between SDEs and RDEs driven by G-Brownian motion. As an application, we introduce the notion of SDEs on a differentiable manifold driven by G-Brownian motion and construct solutions from the RDE point of view by using pathwise localization technique. This is the starting point of developing G-Brownian motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin. The last part of this article is devoted to such construction for a wide and interesting class of G-functions whose invariant group is the orthogonal group. In particular, we establish the generating nonlinear heat equation for such G-Brownian motion on a Riemannian manifold. We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian motion of independent interest.