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Jiangang Ying

Publications and source records attributed to Jiangang Ying.

At least 19 recordsLinked to original sources
arXiv2024

On domination for (non-symmetric) Dirichlet forms

The primary aim of this article is to investigate the domination relationship between two $L^2$-semigroups using probabilistic methods. According to Ouhabaz's domination criterion, the domination of semigroups can be transformed into relationships involving the corresponding Dirichlet forms. Our principal result establishes the equivalence between the domination of Dirichlet forms and the killing transformation of the associated Markov processes, which generalizes and completes the results in \cite{Y962} and \cite{Y96}. Based on this equivalence, we provide a representation of the dominated Dirichlet form using the bivariate Revuz measure associated with the killing transformation and further characterize the sandwiched Dirichlet form within the broader Dirichlet form framework. In particular, our findings apply to the characterization of operators sandwiched between the Dirichlet Laplacian and the Neumann Laplacian. For the local boundary case, we eliminate all technical conditions identified in the literature \cite{AW03} and deliver a complete representation of all sandwiched operators governed by a Robin boundary condition determined by a specific quasi-admissible measure. Additionally, our results offer a comprehensive characterization of related operators in the non-local Robin boundary case, specifically resolving an open problem posed in the literature \cite{OA21}.

math.PR↗
arXiv2023

Fukushima subspaces of quasidiffusions

Quasidiffusions are, by definition, time-changed Brownian motions on certain closed subset of $\mathbb{R}$. They admit an explicit representation of Dirichlet forms in terms of so-called speed measures. The Fukushima subspace of a Dirichlet form means another regular Dichichlet form on the same state space but having a smaller Dirichlet space. In this paper we aim to solve the problem of Fukushima subspaces for quasidiffusions. The main result obtains all Fukushima subspaces and characterizes their structures. In addition, we will also give criteria for the uniqueness of Fukushima subspaces and the existence of minimal Fukushima subspace.

math.PR↗
arXiv2022

On the one-sided boundedness of the local discrepancy of $\{nα\}$-sequences

The main interest of this article is the one-sided boundedness of the local discrepancy of $α\in\mathbb{R}\setminus\mathbb{Q}$ on the interval $(0,c)\subset(0,1)$ defined by \[D_n(α,c)=\sum_{j=1}^n 1_{\{\{jα\}<c\}}-cn.\] We focus on the special case $c\in (0,1)\cap\mathbb{Q}$. Several necessary and sufficient conditions on $α$ for $(D_n(α,c))$ to be one-side bounded are derived. Using these, certain topological properties are given to describe the size of the set \[O_c=\{α\in \irr: (D_n(α,c)) \text{ is one-side bounded}\}.\]

math.NT↗
arXiv2022

Regular subspaces of symmetric stable processes

Roughly speaking, regular subspaces are regular Dirichlet forms that inherit the original forms with smaller domains. In this paper, regular subspaces of 1-dim symmetric $\alpha$-stable processes are considered. The main result is that it admits proper regular subspaces if and only if $\alpha\in [1,2]$. Moreover, for $\alpha\in(1,2)$, the characterization of the regular subspaces is given. General 1-dim symmetric L\'evy processes will also be investigated. It will be shown that whether it has proper regular subspaces is closely related to whether its sample paths have finite variation.

math.PR↗
arXiv2020

Markov chain approximations for one dimensional diffusions

The Markov chain approximation of a one-dimensional symmetric diffusion is investigated in this paper. Given an irreducible reflecting diffusion on a closed interval with scale function $s$ and speed measure $m$, the approximating Markov chains are constructed explicitly through the trace of the Dirichlet form corresponding to the diffusion. One feature of our approach is that it does not require uniform ellipticity on diffusion coefficient of the limit object or uniform regularity on conductances of the approximative Markov chains, as imposed usually in the previous related works.

math.PR↗
arXiv2018

Effective intervals and regular Dirichlet subspaces

It is shown in [10] that a regular and local Dirichlet form on an interval can be represented by so-called effective intervals with scale functions. This paper focuses on how to operate on effective intervals to obtain regular Dirichlet subspaces. The first result is a complete characterization for a Dirichlet form to be a regular subspace of such a Dirichlet form in terms of effective intervals. Then we give an explicit road map how to obtain all regular Dirichlet subspaces from a local and regular Dirichlet form on an interval, by a series of intuitive operations on the effective intervals in the representation above. Finally applying previous results, we shall prove that every regular and local Dirichlet form has a special standard core generated by a continuous and strictly increasing function.

math.PR↗
arXiv2018

On symmetric one-dimensional diffusions

The main purpose of this paper is to explore the structure of local and regular Dirichlet forms associated with symmetric linear diffusions. Let $(\mathcal{E},\mathcal{F})$ be a regular and local Dirichlet form on $L^2(I,m)$, where $I$ is an interval and $m$ is a fully supported Radon measure on $I$. We shall first present a complete representation for $(\mathcal{E},\mathcal{F})$, which shows that $(\mathcal{E},\mathcal{F})$ lives on at most countable disjoint `effective' intervals with corresponding scale function on each interval, and any point outside these intervals is a trap of the linear diffusion. Furthermore, we shall give a necessary and sufficient condition for $C_c^\infty(I)$ being a special standard core of $(\mathcal{E},\mathcal{F})$ and identify the closure of $C_c^\infty(I)$ in $(\mathcal{E},\mathcal{F})$ when $C_c^\infty(I)$ is contained but not necessarily dense in $\mathcal{F}$ relative to the $\mathcal{E}_1$-norm. This paper is partly motivated by a result of [Hamza, 1975], stated in [FOT, Theorem 3.1.6] and provides a different point of view to this theorem. To illustrate our results, many examples are provided.

math.PR↗
arXiv2016

The orthogonal complements of $H^1(\mathbb{R})$ in its regular Dirichlet extensions

Consider the regular Dirichlet extension $(\mathcal{E},\mathcal{F})$ for one-dimensional Brownian motion, that $H^1(\mathbb{R})$ is a subspace of $\mathcal{F}$ and $\mathcal{E}(f,g)=\frac12\mathbf{D}(f,g)$ for $f,g\in H^1(\mathbb{R})$. Both $H^1(\mathbb{R})$ and $\mathcal{F}$ are Hilbert spaces under $\mathcal{E}_α$ and hence there is $α$-orthogonal compliment $\mathcal{G}_α$. We give the explicit expression for functions in $\mathcal{G}_α$ which then can be described by another two spaces. On the two spaces, there is a natural Dirichlet form in the wide sense and by the darning method, their regular representations are given.

math.PR↗
arXiv2016

Regular Dirichlet extensions of one-dimensional Brownian motion

The regular Dirichlet extension is the dual concept of regular Dirichlet subspace. The main purpose of this paper is to characterize all the regular Dirichlet extensions of one-dimensional Brownian motion and to explore their structures. It is shown that every regular Dirichlet extension of one-dimensional Brownian motion may essentially decomposed into at most countable disjoint invariant intervals and an $\mathcal{E}$-polar set relative to this regular Dirichlet extension. On each invariant interval the regular Dirichlet extension is characterized uniquely by a scale function in a given class. To explore the structure of regular Dirichlet extension we apply the idea introduced in [17], we formulate the trace Dirichlet forms and attain the darning process associated with the restriction to each invariant interval of the orthogonal complement of $H^1_\mathrm{e}(\mathbb{R})$ in the extended Dirichlet space of the regular Dirichlet extension. As a result, we find an answer to a long-standing problem whether a pure jump Dirichlet form has proper regular Dirichlet subspaces.

math.PR↗
arXiv2016

On structure of regular Dirichlet subspaces for one-dimensional Brownian motion

The main purpose of this paper is to explore the structure of regular subspaces of 1-dim Brownian motion. As outlined in \cite{FMG} every such regular subspace can be characterized by a measure-dense set $G$. When $G$ is open, $F=G^c$ is the boundary of $G$ and, before leaving $G$, the diffusion associated with the regular subspace is nothing but Brownian motion. Their traces on $F$ still inherit the inclusion relation, in other words, the trace Dirichlet form of regular subspace on $F$ is still a regular subspace of trace Dirichlet form of one-dimensional Brownian motion on $F$. Moreover we have proved that the trace of Brownian motion on $F$ may be decomposed into two part, one is the trace of the regular subspace on $F$, which has only the non-local part and the other comes from the orthogonal complement of the regular subspace, which has only the local part. Actually the orthogonal complement of regular subspace corresponds to a time-changed Brownian motion after a darning transform.

math.PR↗
arXiv2015

Weak convergence of regular Dirichlet subspaces

In this paper we shall prove the weak convergence of the associated diffusion processes of regular subspaces with monotone characteristic sets for a fixed Dirichlet form. More precisely, given a fixed 1-dimensional diffusion process and a sequence of its regular subspaces, if the characteristic sets of regular subspaces are decreasing or increasing, then their associated diffusion processes are weakly convergent to another diffusion process. This is an extended result of [13].

math.PR↗
arXiv2015

Regular subspaces of skew product diffusions

Roughly speaking, the regular subspace of a Dirichlet form is also a regular Dirichlet form on the same state space. It inherits the same form of original Dirichlet form but possesses a smaller domain. What we are concerned in this paper are the regular subspaces of associated Dirichlet forms of skew product diffusions. A skew product diffusion $X$ is a symmetric Markov process on the product state space $E_1\times E_2$ and expressed as \[ X_t=(X^1_t,X^2_{A_t}),\quad t\geq 0, \] where $X^i$ is a symmetric diffusion on $E_i$ for $i=1,2$, and $A$ is a positive continuous additive functional of $X^1$. One of our main results indicates that any skew product type regular subspace of $X$, say \[ Y_t=(Y^1_t,Y^2_{\tilde{A}_t}),\quad t\geq 0, \] can be characterized as follows: the associated smooth measure of $\tilde{A}$ is equal to that of $A$, and $Y^i$ corresponds to a regular subspace of $X^i$ for $i=1,2$. Furthermore, we shall make some discussions on rotationally invariant diffusions on $\mathbf{R}^d\setminus \{0\}$, which are special skew product diffusions on $(0,\infty)\times S^{d-1}$. Our main purpose is to extend a regular subspace of rotationally invariant diffusion on $\mathbf{R}^d\setminus \{0\}$ to a new regular Dirichlet form on $\mathbf{R}^d$.

math.PR↗
arXiv2015

Killing transform on regular Dirichlet subspaces

In this paper, we shall consider the killing transform induced by a multiplicative functional on regular Dirichlet subspaces of a fixed Dirichlet form. Roughly speaking, a regular Dirichlet subspace is a closed subspace with Dirichlet and regular properties of fixed Dirichlet space. By using the killing transforms, our main results indicate that the big jump part of fixed Dirichlet form is not essential for discussing its regular Dirichlet subspaces. This fact is very similar to the status of killing measure when we consider the questions about regular Dirichlet subspaces in [6].

math.PR↗
arXiv2015

Regular subspaces of Dirichlet forms

The regular subspaces of a Dirichlet form are the regular Dirichlet forms that inherit the original form but possess smaller domains. The two problems we are concerned are: (1) the existence of regular subspaces of a fixed Dirichlet form, (2) the characterization of the regular subspaces if exists. In this paper, we will first research the structure of regular subspaces for a fixed Dirichlet form. The main results indicate that the jumping and killing measures of each regular subspace are just equal to that of the original Dirichlet form. By using the independent coupling of Dirichlet forms and some celebrated probabilistic transformations, we will study the existence and characterization of the regular subspaces of local Dirichlet forms.

math.PR↗
arXiv2015

Bivariate Revuz measures and the Feynman-Kac formula on semi-Dirichlet forms

In this paper, we shall first establish the theory of bivariate Revuz correspondence of positive additive functionals under a semi-Dirichlet form, which is associated with a right Markov process $X$ satisfying the sector condition but without duality. We extend most of the classical results about the bivariate Revuz measures under the duality assumptions to the case of semi-Dirichlet forms. As the main results of this paper, we prove that for any exact multiplicative functional $M$ of $X$, the subprocess $X^M$ of $X$ killed by $M$ also satisfies the sector condition and we then characterize the semi-Dirichlet form associated with $X^M$ by using the bivariate Revuz measure, which extends the classical Feynman-Kac formula.

math.PR↗
arXiv2013

Range-Renewal Processes: SLLN, Power Law and Beyonds

Given $n$ samples of a regular discrete distribution $π$, we prove in this article first a serial of SLLNs results (of Dvoretzky and Erdös' type) which implies a typical power law when $π$ is heavy-tailed. Constructing a (random) graph from the ordered $n$ samples, we can establish other laws for the degree-distribution of the graph. The phenomena of small world is also discussed.

math.PR↗
arXiv2009

The uniqueness of symmetrizing measure and linear diffusions

In this short article, we shall study one-dimensional local Dirichlet spaces. One result, which has its independent interest, is to prove that irreducibility implies the uniqueness of symmetrizing measure for right Markov processes. The other result is to give a representation for any 1-dim local, irreducible and regular Dirichlet space and a necessary and sufficient condition for a Dirichlet space to be regular subspace of another Dirichlet space.

math.PR↗