arXiv research

In this paper, we study the gradient estimates of Li-Yau-Hamilton type for positive solutions to both drifting heat equation and the simple nonlinear heat equation problem $u_t-Δu=au\log u, \ \ u>0$ on the compact Riemannian manifold $(M,g)$ of dimension $n$ and with non-negative (Bakry-Emery)-Ricci curvature. Here…

Proves Brownian bridges on manifolds are semimartingales.

The article proves a new entropy formula for surfaces with boundaries.

The paper provides gradient estimates for nonlinear heat-type equations on smooth metric measure spaces.

In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: $u_t=ΔF(u),$ with $F'(u) > 0$, on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equ…

The article derives gradient estimations for semilinear equations on geometric flows.

Chau-Tam-Yu has proved the non-positivity of Perelman's new Li-Yau-Hamilton type expression on noncompact manifolds. In this article, we further prove that $v$ is negative if the Ricci flow is not end up with an Euclidean space.

We give a proof to the Li-Yau-Hamilton type inequality claimed by Perelman on the fundamental solution to the conjugate heat equation. The rest of the paper is devoted to improving the known differential inequalities of Li-Yau-Hamilton type via monotonicity formulae.

We proved a matrix Li-Yau-Hamilton type gradient estimates for the positive solutin of the heat equation on complete Kaehler manifolds with nonnegative bisectional curvature. As a consequence we obtain a comparison theorem for the distance function under this curvature assumption.

The paper provides gradient estimates for a parabolic equation under Finsler geometric flows.

In this paper we study the porous medium equation (PME) coupled with the Ricci flow on complete manifolds with bounded nonnegative curvature operator. In particular, we derive Aronson-Bénilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions to the PME, with a linear forcing term, under the…

Paper proves Harnack inequality for $f$-mean curvature flow.

Optimizes heat equation estimates on noncompact manifolds.

The paper derives new gradient and Hessian estimates for nonlinear parabolic equations.

Study fast diffusion equation under Ricci flow with estimates.

In this paper we study the heat equation (of Hodge-Laplacian) deformation of $(p, p)$-forms on a Kähler manifold. After identifying the condition and establishing that the positivity of a $(p, p)$-form solution is preserved under such an invariant condition we prove the sharp differential Harnack (in the sense of Li-Ya…

The paper improves heat equation estimates under weaker Ricci curvature conditions.

Let $(M,g)$ be a complete non-compact Riemannian manifold with the $m$-dimensional Bakry-Émery Ricci curvature bounded below by a non-positive constant. In this paper, we give a localized Hamilton-type gradient estimate for the positive smooth bounded solutions to the following nonlinear diffusion equation \[ u_t=Δu-\n…

The paper provides gradient estimates for specific evolution equations on metric measure spaces.

Let $(M,g(t))$, $0\le t\le T$, $\partial M\neφ$, be a compact $n$-dimensional manifold, $n\ge 2$, with metric $g(t)$ evolving by the Ricci flow such that the second fundamental form of $\partial M$ with respect to the unit outward normal of $\partial M$ is uniformly bounded below on $\partial M\times [0,T]$. We will pr…

Study harmonic flow of Spin(7)-structures on compact 8-manifolds.

Paper proves Harnack inequalities for Witten Laplacian on manifolds with specific flows.

Let $(M^n, g)$ be a complete Riemannian manifold with $Rc\geq -Kg$, $H(x, y, t)$ is the heat kernel on $M^n$, and $H= (4πt)^{-\frac{n}{2}}e^{-f}$. Nash entropy is defined as $N(H, t)= \int_{M^n} (fH) dμ(x)- \frac{n}{2}$. We studied the asymptotic behavior of $N(H, t)$ and $\frac{\partial}{\partial t}\Big[N(H, t)\Big]$ …

The paper characterizes $(K, m)$-Ricci solitons and flows using $W$-entropy.

New estimators outperform maximum likelihood without hyper-parameter estimation.

Dual Bayesian Affine Estimators for Wiener-type state-space models

New estimator reduces kernel mean estimation error.

Enhances gradient estimates for Hermitian Monge-Ampère equations.

Paper proposes robust estimators for GANs under Wasserstein contamination.

New framework converts offline to online estimation using black-box offline estimators.

Proposes variational autoencoder for efficient MMSE estimation.

Paper presents a robust estimator for density ratio estimation that trims outliers.

Paper improves Fisher information estimation methods.

Estimates local Sobolev constants for manifolds with integral Ricci bounds.

Proposes a robust estimator for RD designs.

New estimator reduces variance in discrete random variables.

New risk-averse estimators uniquely characterize MAP and Wallace-Freeman estimators.

SCOPE estimator improves covariance and precision matrix estimation.

Formalizes robustness for composite estimators using breakdown points.

Obtaining more accurate equity value estimates is the starting point for stock selection, value-based indexing in a noisy market, and beating benchmark indices through tactical style rotation. Unfortunately, discounted cash flow, method of comparables, and fundamental analysis typically yield discrepant valuation estim…

We present a multi-task learning approach to jointly estimate the means of multiple independent data sets. The proposed multi-task averaging (MTA) algorithm results in a convex combination of the single-task maximum likelihood estimates. We derive the optimal minimum risk estimator and the minimax estimator, and show t…

We find an unbiased estimator for MMD variance.

Stochastic volatility modelling of financial processes has become increasingly popular. The proposed models usually contain a stationary volatility process. We will motivate and review several nonparametric methods for estimation of the density of the volatility process. Both models based on discretely sampled continuo…

A new copula estimation method using classification.

Paper improves Li-Yau gradient estimates for heat equation solutions.

This paper reviews SDR methods for multivariate response regression.

TAKDE optimizes kernel density estimation for real-time dynamic processes.

Paper bridges score estimation to parameter and density estimation in DDPMs.