Related papers: Bismut Formula and Gradient Estimates for Dirichle…

In this paper we prove a derivative formula of Bismut-Elworthy-Li's type as well as gradient estimate for stochastic differential equations driven by $\alpha$-stable noises, where $\alpha\in(0,2)$. As an application, the strong Feller…

This paper is concerned with approximations and related discretization error estimates for the normal derivatives of solutions of linear elliptic partial differential equations. In order to illustrate the ideas, we consider the Poisson…

Particle-based variational inference offers a flexible way of approximating complex posterior distributions with a set of particles. In this paper we introduce a new particle-based variational inference method based on the theory of…

By methods of stochastic analysis on Riemannian manifolds, we derive explicit constants $c\_1(D)$ and $c\_2(D)$ for a $d$-dimensional compact Riemannian manifold $D$ with boundary such that $c\_1(D)\sqrt{\lambda}\|\phi\|\_\infty \le…

This study presents new closed-form estimators for the Dirichlet and the Multivariate Gamma distribution families, whose maximum likelihood estimator cannot be explicitly derived. The methodology builds upon the score-adjusted estimators…

We obtain bounded for all $t$ solutions of ordinary differential equations as limits of the solutions of the corresponding Dirichlet problems on $(-L,L)$, with $L \rightarrow \infty$. We derive a priori estimates for the Dirichlet problems,…

We consider distributed smooth nonconvex unconstrained optimization over networks, modeled as a connected graph. We examine the behavior of distributed gradient-based algorithms near strict saddle points. Specifically, we establish that (i)…

We consider the approximation of initial/boundary value problems involving, possibly high-dimensional, dissipative evolution partial differential equations (PDEs) using a deep neural network framework. More specifically, we first propose…

The well-posedness for SDEs with singularity in both space and distribution variables is derived, where the interacting drift term is bounded and Lipschitz continuous under total variation distance and the diffusion term is allowed to be…

The paper presents theorems on the calculation of the index of a singular point and at the infinity of monotone type mappings. These theorems cover basic cases when the principal linear part of a mapping is degenerate. Applications of these…

We consider positive solutions of a fractional Lane-Emden type problem in a bounded domain with Dirichlet conditions. We show that uniqueness and nondegeneracy hold for the asymptotically linear problem in general domains. Furthermore, we…

We present a numerical scheme that can be combined with any fixed boundary finite element based Poisson or Grad-Shafranov solver to compute the first and second partial derivatives of the solution to these equations with the same order of…

This paper considers the problem of understanding the exit time for trajectories of gradient-related first-order methods from saddle neighborhoods under some initial boundary conditions. Given the 'flat' geometry around saddle points,…

We investigate long-time behaviors of empirical measures associated with subordinated Dirichlet diffusion processes on a compact Riemannian manifold $M$ with boundary $\partial M$ to some reference measure, under the quadratic Wasserstein…

In this paper we produce precise large deviation estimates through the lens of mod-Poisson convergence. We apply a general result to various examples from number theory, Dedekind domains and polynomials over finite fields when an element is…

We consider stochastic gradient estimation using only black-box function evaluations, where the function argument lies within a probability simplex. This problem is motivated from gradient-descent optimization procedures in multiple…

In this note we prove sharp lower error bounds for numerical methods for jump-diffusion stochastic differential equations (SDEs) with discontinuous drift. We study the approximation of jump-diffusion SDEs with non-adaptive as well as…

By using Malliavin calculus, Bismut type formulas are established for the Lions derivative of $P_tf(\mu):=\mathbb E f(X_t^\mu)$, where $t>0,$ $ f $ is a bounded measurable function, and $X_t^\mu$ solves a distribution dependent SDE with…

The classical arguments employed when obtaining error estimates of Finite Element (FE) discretisations of elliptic problems lead to more restrictive assumptions on the regularity of the exact solution when applied to non-conforming methods.…

In this paper, we derive $C^2$ estimates for a class of mixed Hessian type equations with Dirichlet boundary condition, and obtain the existence theorem of admissible solutions for the classical Dirichlet problem of these mixed Hessian type…