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Dejun Luo

Publications and source records attributed to Dejun Luo.

64 records · Page 4Linked to original sources
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

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.

math.DG↗
arXiv2010

A note on Gaussian correlation inequalities for nonsymmetric sets

We consider the Gaussian correlation inequality for nonsymmetric convex sets. More precisely, if $A\subset\mathbb{R}^d$ is convex and the origin $0\in A$, then for any ball $B$ centered at the origin, it holds $γ_d(A\cap B)\geq γ_d(A)γ_d(B)$, where $γ_d$ is the standard Gaussian measure on $\mathbb{R}^d$. This generalizes Proposition 1 in [Arch. Rational Mech. Anal. 161 (2002), 257--269].

math.PR↗
arXiv2010

Absolute continuity under flows generated by SDE with measurable drift coefficient

We consider the Itô SDE with non-degenerate diffusion coefficient and 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.

math.PR↗
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↗
arXiv2010

Stochastic differential equations with coefficients in Sobolev spaces

We consider Itô SDE $\d X_t=\sum_{j=1}^m A_j(X_t) \d w_t^j + A_0(X_t) \d t$ on $\R^d$. The diffusion coefficients $A_1,..., A_m$ are supposed to be in the Sobolev space $W_\text{loc}^{1,p} (\R^d)$ with $p>d$, and to have linear growth; for the drift coefficient $A_0$, we consider two cases: (i) $A_0$ is continuous whose distributional divergence $δ(A_0)$ w.r.t. the Gaussian measure $γ_d$ exists, (ii) $A_0$ has the Sobolev regularity $W_\text{loc}^{1,p'}$ for some $p'>1$. Assume $\int_{\R^d} \exp\big[λ_0\bigl(|δ(A_0)| + \sum_{j=1}^m (|δ(A_j)|^2 +|\nabla A_j|^2)\bigr)\big] \dγ_d<+\infty$ for some $λ_0>0$, in the case (i), if the pathwise uniqueness of solutions holds, then the push-forward $(X_t)_# γ_d$ admits a density with respect to $γ_d$. In particular, if the coefficients are bounded Lipschitz continuous, then $X_t$ leaves the Lebesgue measure $\Leb_d$ quasi-invariant. In the case (ii), we develop a method used by G. Crippa and C. De Lellis for ODE and implemented by X. Zhang for SDE, to establish the existence and uniqueness of stochastic flow of maps.

math.PR↗