Dejun Luo

WELL-POSEDNESS OF FOKKER–PLANCK TYPE EQUATIONS ON THE WIENER SPACE

We consider Fokker–Planck type equations on the abstract Wiener space. Under the assumptions that the coefficients have a certain Sobolev regularity and they, together with their gradients and divergences, are exponentially integrable, we establish the existence of solutions to these equations, based on the estimates for solutions to Fokker–Planck equations in the finite-dimensional case. Moreover, the solution is unique if it belongs to the first-order Sobolev space.

Asymptotic estimates on the time derivative of entropy on a Riemannian manifold

We consider the entropy of the solution to the heat equation on a Riemannian manifold. When the manifold is compact, we provide two estimates on the rate of change of the entropy in terms of the lower bound on the Ricci curvature and the spectral gap respectively. Our explicit computation for the three dimensional hyperbolic space shows that the time derivative of the entropy is asymptotically bounded by two positive constants.

Quasi-Invariant Flow Generated by Stratonovich SDE with BV Drift Coefficient

We generalize the results of Ambrosio [1] on the existence, uniqueness, and stability of regular Lagrangian flows of ordinary differential equations to Stratonovich stochastic differential equations with BV drift coefficients. We then construct an explicit solution to the corresponding stochastic transport equation in terms of the stochastic flow. The approximate differentiability of the flow is also studied when the drift is a Sobolev vector field.

Generalized stochastic flow associated to the Itô SDE with partially Sobolev coefficients and its application

We consider the It\^o 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 It\^o SDE with Sobolev coefficients.

Absolute continuity under flows generated by SDE with measurable drift coefficients

We consider the Ito SDE with a non-degenerate diffusion coefficient and a measurable drift coefficient. Under the condition that the gradient of the diffusion coefficient and the divergences of the diffusion and drift coefficients are exponentially integrable with respect to the Gaussian measure, we show that the stochastic flow leaves the reference measure absolutely continuous.

A note on Gaussian correlation inequalities for nonsymmetric sets

We consider the Gaussian correlation inequality for nonsymmetric convex sets. More precisely, if A⊂Rd is convex and the origin 0∈A, then for any ball B centered at the origin, it holds γd(A∩B)≥γd(A)γd(B), where γd is the standard Gaussian measure on Rd. This generalizes Proposition 1 in [Cordero-Erausquin, Dario, 2002. Some applications of mass transport to Gaussian-type inequalities. Arch. Ration. Mech. Anal. 161, 257–269].

The Itô SDEs and Fokker–Planck equations with Osgood and Sobolev coefficients

Abstract We study the degenerate Itô SDE on whose drift coefficient only fulfills a mixed Osgood and Sobolev regularity. Under suitable assumptions on the gradient of the diffusion coefficient and on the divergence of the drift coefficient, we prove the existence and uniqueness of generalized stochastic flows associated to such equations. We also prove the uniqueness of solutions to the corresponding Fokker–Planck equation by using the probabilistic method.

Constantin and Iyer’s Representation Formula for the Navier–Stokes Equations on Manifolds

The purpose of this paper is to establish a probabilistic representation formula for the Navier–Stokes equations on compact Riemannian manifolds. Such a formula has been provided by Constantin and Iyer in the flat case of ℝn or of Tn. On a Riemannian manifold, however, there are several different choices of Laplacian operators acting on vector fields. In this paper, we shall use the de Rham–Hodge Laplacian operator which seems more relevant to the probabilistic setting, and adopt Elworthy–Le Jan–Li’s idea to decompose it as a sum of the square of Lie derivatives.

A characterization of the rate of change ofФ-entropy via an integral form curvature-dimension condition

Abstract Let M be a compact Riemannian manifold without boundary and V : M → ℝ a smooth function. Denote by Pt and dμ = eV dx the semigroup and symmetric measure of the second order differential operator L = ∆ + ∇V · ∇. For some suitable convex function Ф :𝓙 → ℝ defined on an interval 𝓙, we consider the Ф-entropy of Ptf (with respect to μ) for any f ∈ C∞(M, 𝓙). We show that an integral form curvature-dimension condition is equivalent to an estimate on the rate of change of the Ф-entropy. We also generalize this result to bounded smooth domains of a complete Riemannian manifold.