arXiv subjects

Huaiqian Li

Publications and source records attributed to Huaiqian Li.

29 records · Page 2Linked to original sources

arXiv2018

Littlewood--Paley--Stein Estimates for Non-local Dirichlet Forms

We obtain the boundedness in $L^p$ spaces for all $1<p<\infty$ of the so-called vertical Littlewood--Paley functions for non-local Dirichlet forms in the metric measure space under some mild assumptions. For $1<p\le 2$, the pseudo-gradient is introduced to overcome the difficulty that chain rules are not available for non-local operators, and then the Mosco convergence is used to pave the way from the finite jumping kernel case to the general case, while for $2\le p<\infty$, the Burkholder--Davis--Gundy inequality is effectively applied. The former method is analytic and the latter one is probabilistic. The results extend those ones for pure jump symmetric Lévy processes in Euclidean spaces.

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↗

arXiv2016

Sharp Heat Kernel Bounds and Entropy in Metric Measure Spaces

We establish sharp upper and lower bounds of Gaussian type for the heat kernel in the metric measure space satisfying $\RCD(0,N)$ ( equivalently, $\RCD^\ast(0,N)$) condition with $N\in \mathbb{N}\setminus\{1\}$ and having maximum volume growth, and then show its application on the large-time asymptotics of the heat kernel, sharp bounds on the (minimal) Green function, and above all, the large-time asymptotics of the Perelman entropy and the Nash entropy, where for the former the monotonicity of the Perelman entropy is proved. The results generalize the corresponding ones in Riemannian manifolds and also in metric measure spaces obtained recently by the author with R. Jiang and H. Zhang in \cite{JLZ2014}.

math.PR↗

arXiv2015

Lower Bound Estimates for The First Eigenvalue of The Weighted $p$-Laplacian on Smooth Metric Measure Spaces

New lower bounds of the first nonzero eigenvalue of the weighted $p$-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the $m$-Bakry--Émery Ricci curvature, the Escober--Lichnerowicz--Reilly type estimates are proved; under the assumption of nonnegative $\infty$-Bakry--Émery Ricci curvature and the $m$-Bakry--Émery Ricci curvature bounded from below by a non-positive constant, the Li--Yau type lower bound estimates are given. The weighted $p$-Bochner formula and the weighted $p$-Reilly formula are derived as the key tools for the establishment of the above results.

math.DG↗

arXiv2015

Heat Kernel Bounds on Metric Measure Spaces and Some Applications

Let $(X,d,μ)$ be a $RCD^\ast(K, N)$ space with $K\in \mathbb{R}$ and $N\in [1,\infty]$. For $N\in [1,\infty)$, we derive the upper and lower bounds of the heat kernel on $(X,d,μ)$ by applying the parabolic Harnack inequality and the comparison principle, and then sharp bounds for its gradient, which are also sharp in time. When $N=\infty$, we also establish a sharp upper bound of the heat kernel by using the dimension free Harnack inequality. For applications, we study the large time behavior of the heat kernel, the stability of solutions to the heat equation, and show the $L^p$ boundedness of (local) Riesz transforms.

math.MG↗

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

Dimension free Harnack inequalities on $\RCD(K, \infty)$ spaces

The dimension free Harnack inequality for the heat semigroup is established on the $\RCD(K,\infty)$ space, which is a non-smooth metric measure space having the Ricci curvature bounded from below in the sense of Lott-Sturm-Villani plus the Cheeger energy being quadratic. As its applications, the heat semigroup entropy-cost inequality and contractivity properties of the semigroup are studied, and a strong enough Gaussian concentration implying the log-Sobolev inequality is also shown as a generalization of the one on the smooth Riemannian manifold.

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↗

arXiv2013

The log-Sobolev inequality for the ground state of a Schrödinger operator on bounded convex domains

We consider the ground state $ϕ_0$ of the Schrödinger operator $L=-Δ+V$ on the bounded convex domain $Ω\subset\R^n$, satisfying the Dirichlet boundary condition. Assume that $V\in C^1(Ω)$ and it admits an even function $\tilde V\in C^1([-D/2,D/2])$ as its modulus of convexity, where $D$ is the diameter of $Ω$. If the first Dirichlet eigenvalue $\tildeλ_0$ of $-\frac{\d^2}{\d t^2}+\tilde V$ on the interval $[-D/2,D/2]$ satisfies $\tildeλ_0>\tilde V(0)$, then the measure $\dμ=ϕ_0 \d x$ satisfies the log-Sobolev inequality on $Ω$ with the constant $\tildeλ_0-\tilde V(0)$. In particular, if $V$ is convex, then the constant is explicitly given by $\frac{π^2}{D^2}$.

math.PR↗

arXiv2013

A unified treatment of ODEs under Osgood and Sobolev type conditions

In this paper we present a unified treatment for the ordinary differential equations under the Osgood and Sobolev type conditions, following Crippa and de Lellis's direct method. More precisely, we prove the existence, uniqueness and regularity of the DiPerna-Lions flow generated by a vector field which is "almost everywhere Osgood continuous".

math.CA↗

arXiv2010

Quasi-invariant flow generated by Stratonovich SDE with BV drift coefficients

We generalize the results of Ambrosio [Invent. Math. 158 (2004), 227--260] on the existence, uniqueness and stability of regular Lagrangian flows of ordinary differential equations to Stratonovich stochastic differential equations with BV drift coefficients. Then we 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.

math.PR↗