Related papers: Class of smooth functions in Dirichlet spaces

We consider functional differential equations(FDEs) which are perturbations of smooth ordinary differential equations(ODEs). The FDE can involve multiple state-dependent delays or distributed delays (forward or backward). We show that,…

We consider decompositions of processes of the form $Y=f(t,X_t)$ where $X$ is a semimartingale. The function $f$ is not required to be differentiable, so It\^{o}'s lemma does not apply. In the case where $f(t,x)$ is independent of $t$, it…

Two of the authors have defined the class $ WDC(M)$ as the class of all subsets of a smooth manifold $M$ that may be expressed in local coordinates as certain sublevel sets of DC (differences of convex) functions. If $M$ is Riemanian and…

Let $\mathcal{A}$ denote the class of analytic functions $f$ on the unit disk $\mathbb{D}=\{z\in\mathbb{C} : |z|<1\}$, normalized by $f(0)=0$ and $f^{\prime}(0)=1$. For $-\pi/2<\alpha<\pi/2$, let $\mathcal{S}_{\alpha}$ be the subclass of…

We provide two methods for constructing smooth bump functions and for smoothly cutting off smooth functions on fractals, one using a probabilistic approach and sub-Gaussian estimates for the heat operator, and the other using the analytic…

We systematically develop general tools to apply Fukushima's absolute continuity condition. These tools comprise methods to obtain a Hunt process on a locally compact separable metric state space whose transition function has a density…

In this paper, we discuss the 2D convection-diffusion-reaction equation with variable smooth coefficients and the Dirichlet boundary condition on a complicated, thin, and curved domain. We propose the fourth-order compact FDM at every grid…

Three dimensional continuous and discrete Fourier-like transforms, based on the three simple and four semisimple compact Lie groups of rank 3, are presented. For each simple Lie group, there are three families of special functions ($C$-,…

The aim of the present work is to show how, using the differential calculus associated to Dirichlet forms, it is possible to construct Fredholm modules on post critically finite fractals by regular harmonic structures. The modules are…

In this paper we provide a different approach for existence of the variational solutions of the gradient flows associated to functionals on Sobolev spaces studied in \cite{BDDMS20}. The crucial condition is the convexity of the functional…

We develop sufficient analytic conditions for conservativeness of non-sectorial perturbations of symmetric Dirichlet forms which can be represented through a carr\'e du champ on a locally compact separable metric space. These form an…

General theorems on the closability and quasi-regularity of non-local Markovian symmetric forms on probability spaces $(S, {\cal B}(S), \mu)$, with $S$ Fr{\'e}chet spaces such that $S \subset {\mathbb R}^{\mathbb N}$, ${\cal B}(S)$ is the…

On a 6-dimensional, conformal, oriented, compact manifold $M$ without boundary, we compute a whole family of differential forms $\Omega_6(f,h)$ of order 6, with $f,h \in C^\infty(M).$ Each of these forms will be symmetric on $f,$ and $h,$…

The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of "weak Dirichlet process" in this context. Such a process $\X$,…

This paper provides a comprehensive Sobolev regularity theory for the Dirichlet problem of stochastic partial differential equations in $C^{1,\sigma}$ open sets. We consider substantially large classes of nonlocal operators and generalized…

Let $L$ be the distinguished Laplacian on the Iwasawa $AN$ group associated with a semisimple Lie group $G$. Assume $F$ is a Borel function on $\mathbb{R}^+$. We give a condition on $F$ such that the kernels of the functions $F(L)$ are…

We establish a necessary and sufficient condition on a continuous function on $[-1,1]$ under which the family of functions on the unit sphere $\mathbb{S}^{d-1}$ constructed in the described manner is fundamental in $C(\mathbb{S}^{d-1})$. In…

The goal of this note is to generalize Thurston's Topological Characterization of Rational Functions to the setting when both the covering degree and the set of marked points are infinite. A relevant class of branched coverings are…

It is an open question whether solutions of the Einstein-Euler equations are smooth enough to admit locally inertial coordinates at points of shock wave interaction, or whether "regularity singularities" can exist at such points. The term…

This is the first of a series of papers on the interior regularity of fully nonlinear degenerate elliptic equations. We consider a stochastic optimal control problem in which the diffusion coefficients, drift coefficients and discount…