Related papers: Nash inequality for Diffusion Processes Associated…
Dirichlet distribution and Dirichlet process as its infinite dimensional generalization are primarily used conjugate prior of categorical and multinomial distributions in Bayesian statistics. Extensions have been proposed to broaden…
We study the homogeneous Cauchy-Dirichlet Problem (CDP) for a nonlinear and nonlocal diffusion equation of singular type of the form $\partial_t u =-\mathcal{L} u^m$ posed on a bounded Euclidean domain $\Omega\subset\mathbb{R}^N$ with…
We study the escape rate of diffusion process with two approaches. We first give an upper rate function for the diffusion process associated with a symmetric, strongly local regular Dirichlet form. The upper rate function is in terms of the…
We study nonlocal Dirichlet energies associated with a class of nonlocal diffusion models on a bounded domain subject to the conventional local Dirichlet boundary condition. The goal of this paper is to give a general framework to correctly…
A stochastic dynamics $({\bf X}(t))_{t\ge0}$ of a classical continuous system is a stochastic process which takes values in the space $\Gamma$ of all locally finite subsets (configurations) in $\Bbb R$ and which has a Gibbs measure $\mu$ as…
It is well known that Nash-type inequalities for symmetric Dirichlet forms are equivalent to on-diagonal heat kernel upper bounds for the associated symmetric Markov semigroups. In this paper, we show that both imply (and hence are…
Assessing homogeneity of distributions is an old problem that has received considerable attention, especially in the nonparametric Bayesian literature. To this effect, we propose the semi-hierarchical Dirichlet process, a novel hierarchical…
The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N coupled stochastic variables with the Dirichlet distribution as its asymptotic solution. To ensure a bounded…
We study the distributions of the random Dirichlet series with parameters $(s, \beta)$ defined by $$ S=\sum_{n=1}^{\infty}\frac{I_n}{n^s}, $$ where $(I_n)$ is a sequence of independent Bernoulli random variables, $I_n$ taking value $1$ with…
We establish a precise connection between two elliptic quasilinear problems with Dirichlet data in a bounded domain of $\mathbb{R}^{N}.$ The first one, of the form \[ -\Delta_{p}u=\beta(u)| \nabla u| ^{p}+\lambda f(x)+\alpha, \] involves a…
The dimension-free Harnack inequality and uniform heat kernel upper/lower bounds are derived for a class of infinite-dimensional GEM processes, which was introduced in \cite{FW} to simulate the two-parameter GEM distributions. In…
We investigate the distribution of values of cubic Dirichlet $L$-functions at $s=1$. Following ideas of Granville and Soundararajan for quadratic $L$-functions, we model the distribution of $L(1,\chi)$ by the distribution of random Euler…
The diffusion of finite-size hard-core interacting particles in two- or three-dimensional confined domains is considered in the limit that the confinement dimensions become comparable to the particle's dimensions. The result is a nonlinear…
In this paper, we will consider the problem that how far from Hunt's hypothesis (H) to symmetrization for a general 1-dimensional diffusion. A characterization of (H) involving the classification of points for this diffusion will be first…
We investigate the Dirichlet problem of the two dimensional Lagrangian mean curvature equation in a bounded domain. Infinitely many $C^{1, \alpha} (\alpha\in (0,\frac{1}{5}))$ very weak solutions are built through Nash-Kuiper construction.…
We study diffusion processes driven by a Brownian motion with regular drift in a finite dimension setting. The drift has two components on different time scales, a fast conservative component and a slow dissipative component. Using the…
Application of fractional calculus to the description of anomalous diffusion and relaxation processes in complex media provided one of the most impressive impulses to the development of statistical physics during the last decade. In…
Let $x \in [0,1)$ be an irrational number with continued fraction expansion $[a_1(x),a_2(x), \cdots,a_n(x),\cdots]$ and $q_n(x)$ be the denominator of its $n$-th convergent. We establish, for any $\alpha,\beta$ in $[0,+\infty]$, the…
We consider the Dirichlet problem for second-order linear elliptic equations in divergence form \begin{equation*} -\mathrm{div }(A\nabla u)+\mathbf{b} \cdot \nabla u+\lambda u=f+\mathrm{div } \mathbf{F}\quad \text{in }…
In this paper, a diffusion-aggregation equation with delta potential is introduced. Based on the global existence and uniform estimates of solutions to the diffusion-aggregation equation, we also provide the rigorous derivation from a…