Conformal Perturbation of Heat Kernels with applications

Let $(M, g)$ be a smooth n-dimensional Riemannian manifold for $n\ge 2$. Consider the conformal perturbation $\tilde{g}=h g$ where $h$ is a smooth bounded positive function on $M$. Denote by $\tilde{p}_t(x,y)$ the heat kernel of manifolds $(M, \tilde{g})$. In this paper, we derive the upper bounds and gradient estimates of $\tilde{p}_t(x,y)$

Uniform Complex Time Heat Kernel Estimates Without Gaussian Bounds

In this paper, first we consider the uniform complex time heat kernel estimates of $e^{-z(-\Delta)^{\frac{\alpha}{2}}}$ for $\alpha>0, z\in \mathbb{C}^+$. When $\frac{\alpha}{2}$ is not an integer, generally the heat kernel doest not have the Gaussian upper bounds for real time. Thus the Phragm\'en-Lindel\"of methods fail to give the uniform complex time estimates. Instead, our first result gives the asymptotic estimates for $P(z, x)$ as $z$ tending to the imaginary axis. Then we prove the uniform complex time heat kernel estimates. Finally we also show the uniform estimates of analytic semigroup generated by $H=(-\Delta)^{\frac{\alpha}{2}}+V$ where $V$ belongs to higher order Kato class

Estimating the Spatial Distribution of Groundwater Demand In the Texas High Plains

Developing groundwater management plans requires a good understanding of the interdependence of groundwater hydrology and producer water use behavior. While state-of-the-art groundwater models require water demand data at highly disaggregated levels, the lack of producer water use data has held up the progress to meet that need. This paper proposes an econometric framework that links county-level crop acreage data to well-level hydrologic data to produce heterogeneous patterns of crop choice and irrigation practices within a county. Together with agronomic data on irrigation water requirements of various crops and irrigation practices, this model permits estimation of the water demand distribution within a county. We apply this model to a panel of 16 counties in the Southern Texas High Plains from 1972 to 2000. The results obtained not only are consistent with those from the traditional multinomial logit land use model, but also indicate the presence of large intra- and inter-county heterogeneity in producer water use behavior.Discrete Choice Model, Random-coefficients Discrete Choice Model, Crop Choice, BLP, Groundwater, Texas High Plains, Ogallala Aquifer, Crop Production/Industries, Resource /Energy Economics and Policy,

Variational Dependent Multi-output Gaussian Process Dynamical Systems

This paper presents a dependent multi-output Gaussian process (GP) for modeling complex dynamical systems. The outputs are dependent in this model, which is largely different from previous GP dynamical systems. We adopt convolved multi-output GPs to model the outputs, which are provided with a flexible multi-output covariance function. We adapt the variational inference method with inducing points for approximate posterior inference of latent variables. Conjugate gradient based optimization is used to solve parameters involved. Besides the temporal dependency, the proposed model also captures the dependency among outputs in complex dynamical systems. We evaluate the model on both synthetic and real-world data, and encouraging results are observed