Convergence of densities of some functionals of Gaussian processes

The aim of this paper is to establish the uniform convergence of the densities of a sequence of random variables, which are functionals of an underlying Gaussian process, to a normal density. Precise estimates for the uniform distance are derived by using the techniques of Malliavin calculus, combined with Stein's method for normal approximation. We need to assume some non-degeneracy conditions. First, the study is focused on random variables in a fixed Wiener chaos, and later, the results are extended to the uniform convergence of the derivatives of the densities and to the case of random vectors in some fixed chaos, which are uniformly non-degenerate in the sense of Malliavin calculus. Explicit upper bounds for the uniform norm are obtained for random variables in the second Wiener chaos, and an application to the convergence of densities of the least square estimator for the drift parameter in Ornstein-Uhlenbeck processes is discussed

Feynman--Kac formula for the heat equation driven by fractional noise with Hurst parameter H<1/2H<1/2

In this paper, a Feynman-Kac formula is established for stochastic partial differential equation driven by Gaussian noise which is, with respect to time, a fractional Brownian motion with Hurst parameter $H<1/2$. To establish such a formula, we introduce and study a nonlinear stochastic integral from the given Gaussian noise. To show the Feynman--Kac integral exists, one still needs to show the exponential integrability of nonlinear stochastic integral. Then, the approach of approximation with techniques from Malliavin calculus is used to show that the Feynman-Kac integral is the weak solution to the stochastic partial differential equation.Comment: Published in at http://dx.doi.org/10.1214/11-AOP649 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Improving Routing Efficiency through Intermediate Target Based Geographic Routing

The greedy strategy of geographical routing may cause the local minimum problem when there is a hole in the routing area. It depends on other strategies such as perimeter routing to find a detour path, which can be long and result in inefficiency of the routing protocol. In this paper, we propose a new approach called Intermediate Target based Geographic Routing (ITGR) to solve the long detour path problem. The basic idea is to use previous experience to determine the destination areas that are shaded by the holes. The novelty of the approach is that a single forwarding path can be used to determine a shaded area that may cover many destination nodes. We design an efficient method for the source to find out whether a destination node belongs to a shaded area. The source then selects an intermediate node as the tentative target and greedily forwards packets to it, which in turn forwards the packet to the final destination by greedy routing. ITGR can combine multiple shaded areas to improve the efficiency of representation and routing. We perform simulations and demonstrate that ITGR significantly reduces the routing path length, compared with existing geographic routing protocols

Hexapod Coloron at the LHC

Instead of the usual dijet decay, the coloron may mainly decay into its own "Higgs bosons", which subsequently decay into many jets. This is a general feature of the renormalizable coloron model, where the corresponding "Higgs bosons" are a color-octet $\Theta$ and a color-singlet $\phi_I$. In this paper, we perform a detailed collider study for the signature of $pp \rightarrow G' \rightarrow (\Theta \rightarrow gg) (\phi_I \rightarrow gg q\bar{q})$ with the coloron $G'$ as a six-jet resonance. For a light $\phi_I$ below around 0.5 TeV, it may be boosted and behave as a four-prong fat jet. We also develop a jet-substructure-based search strategy to cover this boosted $\phi_I$ case. Independent of whether $\phi_I$ is boosted or not, the 13 TeV LHC with 100 fb$^{-1}$ has great discovery potential for a coloron with the mass sensitivity up to 5 TeV.Comment: 18 pages, 10 figure

Joint state-parameter estimation of a nonlinear stochastic energy balance model from sparse noisy data

While nonlinear stochastic partial differential equations arise naturally in spatiotemporal modeling, inference for such systems often faces two major challenges: sparse noisy data and ill-posedness of the inverse problem of parameter estimation. To overcome the challenges, we introduce a strongly regularized posterior by normalizing the likelihood and by imposing physical constraints through priors of the parameters and states. We investigate joint parameter-state estimation by the regularized posterior in a physically motivated nonlinear stochastic energy balance model (SEBM) for paleoclimate reconstruction. The high-dimensional posterior is sampled by a particle Gibbs sampler that combines MCMC with an optimal particle filter exploiting the structure of the SEBM. In tests using either Gaussian or uniform priors based on the physical range of parameters, the regularized posteriors overcome the ill-posedness and lead to samples within physical ranges, quantifying the uncertainty in estimation. Due to the ill-posedness and the regularization, the posterior of parameters presents a relatively large uncertainty, and consequently, the maximum of the posterior, which is the minimizer in a variational approach, can have a large variation. In contrast, the posterior of states generally concentrates near the truth, substantially filtering out observation noise and reducing uncertainty in the unconstrained SEBM