A fractional generalization of the Dirichlet distribution and related distributions

带Dirichlet边界分数噪声的二维Navier-Stokes方程的全局适定性

Global well-posedness of 2D Navier-Stokes with Dirichlet boundary fractional noise

本文研究带非齐次Dirichlet型分数边界噪声的二维Navier-Stokes方程，证明其全局适定性与内部正则性，该模型为描述海洋切片相对运动的随机Couette流，所用高斯噪声Hurst参数大于3/4。

BV functions and fractional Laplacians on Dirichlet spaces

We study $L^p$ Besov critical exponents and isoperimetric and Sobolev inequalities associated with fractional Laplacians on metric measure spaces. The main tool is the theory of heat semigroup based Besov classes in Dirichlet spaces that was introduced by the authors in previous works.

A microscopic model for a one parameter class of fractional laplacians with dirichlet boundary conditions

We prove the hydrodynamic limit for the symmetric exclusion process with long jumps given by a mean zero probability transition rate with infinite variance and in contact with infinitely many reservoirs with density $α$ at the left of the system and $β$ at the right of the system. The strength of the reservoirs is ruled by $κ$N --$θ$ > 0. Here N is the size of the system, $κ$ > 0 and $θ$ $\in$. Our results are valid for $θ$ $\le$ 0. For $θ$ = 0, we obtain a collection of fractional reaction-diffusion equations indexed by the parameter $κ$ and with Dirichlet boundary conditions. Their solutions…

Nonstandard Representation of the Dirichlet Form

The Dirichlet form is a generalization of the Laplacian, heavily used in the study of many diffusion-like processes. In this paper we present a nonstandard representation theorem for the Dirichlet form, showing that the usual Dirichlet form can be well-approximated by a hyperfinite sum. One of the main motivations for such a result is to provide a tool for directly translating results about Dirichlet forms on finite or countable state spaces to results on more general state spaces, without having to translate the details of the proofs. As an application, we prove a generalization of a well-kno…

Fractional equations via convergence of forms

We relate the convergence of time-changed processes driven by fractional equations to the convergence of corresponding Dirichlet forms. The fractional equations we dealt with are obtained by considering a general fractional operator in time.

A new approach for the fractional Laplacian via deep neural networks

The fractional Laplacian has been strongly studied during past decades. In this paper we present a different approach for the associated Dirichlet problem, using recent deep learning techniques. In fact, intensively PDEs with a stochastic representation have been understood via neural networks, overcoming the so-called curse of dimensionality. Among these equations one can find parabolic ones in $\mathbb{R}^d$ and elliptic in a bounded domain $D \subset \mathbb{R}^d$. In this paper we consider the Dirichlet problem for the fractional Laplacian with exponent $α\in (1,2)$. We show that its solut…

A Bivariate Beta Distribution with Arbitrary Beta Marginals and its Generalization to a Correlated Dirichlet Distribution

We discuss a bivariate beta distribution that can model arbitrary beta-distributed marginals with a positive correlation. The distribution is constructed from six independent gamma-distributed random variates. We show how the parameters of the distribution can be fit to data using moment matching. Previous work used an approximate and sometimes inaccurate method to compute the covariance. Here, we derive all product moments and the exact covariance, which can easily be computed numerically. The bivariate case can be generalized to a multivariate distribution with arbitrary beta-distributed mar…

椭圆和抛物型边值问题的随机解法：谱分数阶拉普拉斯算子

Stochastic Solution of Elliptic and Parabolic Boundary Value Problems for the Spectral Fractional Laplacian

本文提出并实现了涉及非零狄利克雷边界条件的谱分数阶拉普拉斯算子的随机解法（或费曼-科茨公式），通过抽象柯西问题和巴拉克里什南的分数次幂理论，证明了该算子是合适费曼-科茨半群的生成元，并通过特征函数展开获得了抛物型问题的正则性和稳态收敛性质，从而建立了狄利克雷边值问题的随机解公式，为高效并行求解提供了新方法。