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Dejun Luo

Publications and source records attributed to Dejun Luo.

At least 55 records · Page 3Linked to original sources
arXiv2019

Regularization by noise for the point vortex model of mSQG equations

We consider the point vortex model corresponding to the modified Surface Quasi-Geostrophic (mSQG) equations on the two dimensional torus. It is known that this model is well posed for almost every initial conditions. We show that, when the system is perturbed by a certain space-dependent noise, it admits a unique global solution for any initial configuration. We also present an explicit example for the deterministic system where three different point vortices collapse.

math.PR↗
arXiv2019

Energy conditional measures and 2D turbulence

We show that the invariant measure of point vortices, when conditioning the Hamiltonian to a finite interval, converges weakly to the enstrophy measure by conditioning the renormalized energy to the same interval. We also prove the existence of solutions to 2D Euler equations having the energy conditional measure as invariant measure. Some heuristic discussions and numerical simulations are presented in the last section.

math-ph↗
arXiv2018

Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises

We establish the exponential convergence with respect to the $L^1$-Wasserstein distance and the total variation for the semigroup corresponding to the stochastic differential equation (SDE) $$d X_t=d Z_t+b(X_t)\,d t,$$ where $(Z_t)_{t\ge0}$ is a pure jump Lévy process whose Lévy measure $ν$ fulfills $$ \inf_{x\in \R^d, |x|\le κ_0} [ν\wedge (δ_x \ast ν)]( \R^d)>0$$ for some constant $κ_0>0$, and the drift term $b$ satisfies that for any $x,y\in \R^d$, $$\langle b(x)-b(y),x-y\rangle\le \begin{cases} Φ_1(|x-y|)|x-y|,& |x-y|\le l_0; -K_2|x-y|^2,& |x-y|> l_0 \end{cases}$$ with some positive constants $K_2, l_0$ and positive measurable function $Φ_1$. The method is based on the refined basic coupling for Lévy jump processes. As a byproduct, we obtain sufficient conditions for the strong ergodicity of the process $(X_t)_{t\ge0}$.

math.PR↗
arXiv2018

Kolmogorov equations associated to the stochastic 2D Euler equations

The Kolmogorov equation associated to a stochastic 2D Euler equations with transport type noise and random initial conditions is studied by a direct approach, based on Fourier analysis, Galerkin approximation and Wiener chaos methods. The method allows us to generalize previous results and to understand the role of the regularity of the noise, in relation to a limiting value of roughness.

math.PR↗
arXiv2018

Euler-Lagrangian approach to 3D stochastic Euler equations

3D stochastic Euler equations with a special form of multiplicative noise are considered. A Constantin-Iyer type representation in Euler-Lagrangian form is given, based on stochastic characteristics. Local existence and uniqueness of solutions in suitable Hoelder spaces is proved from the Euler-Lagrangian formulation.

math.PR↗
arXiv2017

$ρ$-white noise solution to 2D stochastic Euler equations

A stochastic version of 2D Euler equations with transport type noise in the vorticity is considered, in the framework of Albeverio--Cruzeiro theory [1] where the equation is considered with random initial conditions related to the so called enstrophy measure. The equation is studied by an approximation scheme based on random point vortices. Stochastic processes solving the Euler equations are constructed and their density with respect to the enstrophy measure is proved to satisfy a continuity equation in weak form. Relevant in comparison with the case without noise is the fact that here we prove a gradient type estimate for the density. Although we cannot prove uniqueness for the continuity equation, we discuss how the gradient type estimate may be related to this open problem.

math.PR↗
arXiv2017

Quantitative stability estimates for Fokker-Planck equations

We consider the Fokker--Planck equations with irregular coefficients. Two different cases are treated: in the degenerate case, the coefficients are assumed to be weakly differentiable, while in the non-degenerate case the drift satisfies only the Ladyzhenskaya--Prodi--Serrin condition. Using Trevisan's superposition principle which represents the solution as the marginal of the solution to the martingale problem of the diffusion operator, we establish quantitative stability estimates for the solutions of Fokker--Planck equations.

math.PR↗
arXiv2017

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 $\mathbb R^n$ or of $\mathbb T^n$. 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.

math.PR↗
arXiv2016

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

We study the degenerated Itô SDE on $\mathbb R^d$ 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.

math.PR↗
arXiv2016

Exponential Convergence in $L^p$-Wasserstein Distance for Diffusion Processes without Uniformly Dissipative Drift

By adopting the coupling by reflection and choosing an auxiliary function which is convex near infinity, we establish the exponential convergence of diffusion semigroups $(P_t)_{t\ge0}$ with respect to the standard $L^p$-Wasserstein distance for all $p\in[1,\infty)$. In particular, we show that for the Itô stochastic differential equation $$\d X_t=\d B_t+b(X_t)\,\d t,$$ if the drift term $b$ satisfies that for any $x,y\in\R^d$, $$\langle b(x)-b(y),x-y\rangle\le \begin{cases} K_1|x-y|^2,& |x-y|\le L; -K_2|x-y|^2,& |x-y|> L \end{cases}$$ holds with some positive constants $K_1$, $K_2$ and $L>0$, then there is a constant $λ:=λ(K_1,K_2,L)>0$ such that for all $p\in[1,\infty)$, $t>0$ and $x,y\in\R^d$, $$W_p(δ_x P_t,δ_y P_t)\leq Ce^{-λt/p} \begin{cases} |x-y|^{1/p}, & \mbox{if } |x-y|\le 1; |x-y|, & \mbox{if } |x-y|> 1. \end{cases}$$ where $C:=C(K_1,K_2,L,p)$ is a positive constant. This improves the main result in \cite{Eberle} where the exponential convergence is only proved for the $L^1$-Wasserstein distance.

math.PR↗
arXiv2015

A probabilistic proof of the fundamental gap conjecture via the coupling by reflection

Let $Ω\subset\mathbb{R}^n$ be a strictly convex domain with smooth boundary and diameter $D$. The fundamental gap conjecture claims that if $V:\barΩ\to\mathbb{R}$ is convex, then the spectral gap of the Schrödinger operator $-Δ+V$ with Dirichlet boundary condition is greater than $\frac{3π^2}{D^2}$. Using analytic methods, Andrews and Clutterbuck recently proved in [J. Amer. Math. Soc. 24 (2011), no. 3, 899--916] a more general spectral gap comparison theorem which implies this conjecture. In the first part of the current work, we shall give an independent probabilistic proof of their result via the coupling by reflection of the diffusion processes. Moreover, we also present in the second part a simpler probabilistic proof of the original conjecture.

math.PR↗
arXiv2015

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

Let $M$ be a compact Riemannian manifold without boundary and $V:M\to \mathbb R$ a smooth function. Denote by $P_t$ and ${\rm d}μ=e^V\,{\rm d} x$ the semigroup and symmetric measure of the second order differential operator $L=Δ+\nabla V\cdot\nabla$. For some suitable convex function $Φ:{\mathcal I}\to\mathbb R$ defined on an interval $\mathcal I$, we consider the $Φ$-entropy of $P_t f$ (with respect to $μ$) for any $f\in C^\infty(M,\mathcal I)$. We show that an integral form curvature-dimension condition is equivalent to an estimate on the rate of change of the $Φ$-entropy. We also generalize this result to bounded smooth domains of a complete Riemannian manifold.

math.DG↗
arXiv2014

Stochastic Lagrangian flows on the group of volume-preserving homeomorphisms of the spheres

We consider stochastic differential equations on the group of volume-preserving homeomorphisms of the sphere $S^d\,(d\geq 2)$. The diffusion part is given by the divergence free eigenvector fields of the Laplacian acting on $L^2$-vector fields, while the drift is some other divergence free vector field. We show that the equation generates a unique flow of measure-preserving homeomorphisms when the drift has first order Sobolev regularity, and derive a formula for the distance between two Lagrangian flows. We also compute the rotation process of two particles on the sphere $S^2$ when they are close to each other.

math.PR↗
arXiv2014

Harnack Inequalities for SDEs with Multiplicative Noise and Non-regular Drift

The log-Harnack inequality and Harnack inequality with powers for semigroups associated to SDEs with non-degenerate diffusion coefficient and non-regular time-dependent drift coefficient are established, based on the recent papers \cite{Flandoli, Zhang11}. We consider two cases in this work: (1) the drift fulfills the LPS-type integrability, and (2) the drift is uniformly Hölder continuous with respect to the spatial variable. Finally, by using explicit heat kernel estimates for the stable process with drift, the Harnack inequality for the stochastic differential equation driven by symmetric stable process is also proved.

math.PR↗
arXiv2014

Uniform Hölder Estimates on Semigroups Generated by Non-Local Operators of Variable Order

We consider the non-local operator of variable order as follows $$Lf(x)= \int_{\R^d\setminus\{0\}}\big(f(x+z)-f(x)-\<\nabla f(x),z\> \I_{\{|z|\le 1\}}\big)\frac{n(x,z)}{|z|^{d+α(x)}}\,dz.$$ Under mild conditions on $α(x)$ and $n(x,z)$, we establish the Hölder regularity for the associated semigroups. The proof is based on the probabilistic coupling method, and it successfully applies to both stable-like processes in the sense of Bass and time-change of symmetric stable processes.

math.PR↗
arXiv2014

Quasi-invariance of the stochastic flow associated to Itô's SDE with singular time-dependent drift

In this paper we consider the Itô SDE $$d X_t=d W_t+b(t,X_t)\,d t, \quad X_0=x\in {\mathbb R}^d,$$ where $W_t$ is a $d$-dimensional standard Wiener process and the drift coefficient $b:[0,T]\times{\mathbb R}^d\to{\mathbb R}^d$ belongs to $L^q(0,T;L^p({\mathbb R}^d))$ with $p\geq 2, q>2$ and $\frac dp +\frac 2q<1$. In 2005, Krylov and Röckner \cite{KR05} proved that the above equation has a unique strong solution $X_t$. Recently it was shown by Fedrizzi and Flandoli \cite{FF13b} that the solution $X_t$ is indeed a stochastic flow of homeomorphisms on ${\mathbb R}^d$. We prove in the present work that the Lebesgue measure is quasi-invariant under the flow $X_t$.

math.PR↗