A Note on Feller Semigroups and Resolvents
Vadim Kostrykin, Jürgen Potthoff, Robert Schrader
给出了由马尔可夫过程生成的半群或预解式为Feller型的各种等价条件。
Various equivalent conditions for a semigroup or a resolvent generated by a Markov process to be of Feller type are given.
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In this paper, we give sufficient conditions for the equivalence between semigroup strong Feller property and resolvent strong Feller property.
Strong Feller Continuity of Feller Processes and Semigroups
本文研究了马尔可夫过程的强Feller性质的两个等价刻画以及相应的子马尔可夫半群的强Feller性质。一个是局部一致的绝对连续性，而另一个使用本地Orlicz超压缩。这些准则推广了许多现有的强Feller连续性的结果，似乎是更自然的Feller过程。通过建立首次出球时间的估计，我们还研究了具有强Feller性质的Feller过程的调和函数的连续性。
The asymptotic strong Feller property does not imply the e-property for Markov-Feller semigroups
T. Szarek [Stud.Math.189（2008），§ 4]讨论了关于马尔可夫半群的两个重要概念：渐近强Feller性质和e-性质之间的关系，断言前一性质蕴涵后一性质。在这篇很短的笔记中，我们纠正了这个问题，展示了一个简单的例子，Markov-Feller半群享有渐近强Feller性质，其中e-性质不满足。(See并对T. Szarek，D.蠕虫[ETDS 32（2012），§ 1]）。此外，我们还给出了一个非常简单的例子--与T. Szarek [Stud.Math.189（2008），§ 4] --表明逆蕴涵也不成立。
Ergodicity and asymptotic stability of Feller semigroups on Polish metric spaces
给出了Polish度量空间上一般Feller半群的遍历性和渐近稳定性的一些精确判据。作为应用，我们简单地讨论了退化随机强迫下的二维Navier-斯托克斯方程。
Perturbation of strong Feller semigroups and well-posedness of semilinear stochastic equations on Banach spaces
证明了强Feller半群的Miyadera-Voigt型扰动定理。利用这一结果，我们证明了半线性随机方程dX（t）= [AX（t）+ F（X（t））]dt + GdW_H（t）在可分巴拿赫空间E上的适定性，假定F是有界可测的，相应的线性方程，即F = 0的方程是适定性的，其转移半群是强Feller的，并满足适当的梯度估计.我们还研究了伴随转移半群的不变测度的存在唯一性。
We investigate the relation between the e-property and the eventual continuity, or called the asymptotic equicontinuity, which is a generalization of the e-property. We prove that, for any discrete-time or strongly continuous continuous-time eventually continuous Markov-Feller semigroup with an ergodic measure, if the interior of the support of the ergodic measure is nonempty, then the e-property is satisfied on the interior of the support. In particular, it implies that, restricted on the support of each ergodic measure, the e-property and the eventual continuity are equivalent for the discre…
In this article we study a class of non-Archimedean pseudodifferential operators whose symbols are negative definite functions. We prove that these operators extend to generators of Feller semigroups. In order to study these operators, we introduce a new class of anisotropic Sobolev spaces, which are the natural domains for the operators considered here. We also study the Cauchy problem for certain pseudodifferential equations.
A limit theorem for certain Feller semigroups, and the distribution of the local time for Brownian motion when it exits an interval
本文从半群理论视角研究一维 Laplace 算子的局部奇异摄动，证明其收敛至零点具 Robin 型边界条件的热半群，并借此提出计算 Brownian motion 离开区间时局部时分布的方法。
The aim of this note is to show, by providing an elementary way to construct counter-examples, that the strong Feller and the joint (space-time) continuity for a semigroup of Markov kernels on a Polish space are not enough to ensure the existence of an associated càdlàg Markov process on the same space. One such simple counter-example is the Brownian semigroup on $\mathbb{R}$ restricted to $\mathbb{R}\setminus \{0\}$, for which it is shown that there is no associated càdlàg Markov process. Using the same idea and results from potential theory we then prove that the analogous result with càdlàg…
In this paper, we consider first order Sobolev spaces with Robin boundary condition on unbounded Lipschitz domains. Hunt processes are associated with these spaces. We prove that the semigroup of these processes are doubly Feller. As a corollary, we provide a condition for semigroups generated by these processes being compact.
Existence and uniqueness of invariant measures for non-Feller Markov semigroups
本文研究非Feller马尔可夫半群不变测度的存在性和唯一性，通过测度论方法证明在自然假设下唯一性成立。
Criteria for asymptotic stability of eventually continuous Markov-Feller semigroups
本文研究 Markov-Feller 半群的渐近稳定性，提出仅需单点最终连续性的收敛与稳定性判据，并用显式例子展示其相较全局条件更易验证。
Ergodicity for eventually continuous Markov--Feller semigroups on Polish spaces
本文研究波兰空间上马尔可夫--费勒半群在极弱正则性条件下的遍历性，证明了Cesàro平均的弱收敛性，刻画了Cesàro最终连续性、e-性质与弱-*平均遍历性的关系，并给出了不变测度存在唯一性准则及遍历分解性质，最后应用于多个非平凡系统。
Feller property and convergence for semigroups of time-changed processes
本文为时间变换过程半群提供了Feller性质的替代条件，并建立了半群与预解式的收敛性，应用于演化方程解、有限时间分布及过程弱收敛。
The comonotone flow of a stochastically monotone Feller process on the real line
本文研究了随机单调Feller过程在实数线上通过保持顺序的半群族扩展的问题，并展示了其在超模序下的唯一性性质及耦合存在的结论。
Neumann semigroup, subgraph convergence, form uniqueness, stochastic completeness and the Feller property
本文研究了带 Neumann 边界条件的导出子图的热核收敛性，建立了所得半群在 $\ell^2$ 中向 Neumann 半群的收敛性，揭示了其与 Dirichlet 半群收敛及随机完备性的等价关系，并通过广义解探讨了 Feller 性质及其应用。
Transition Densities and Traces for Invariant Feller Processes on Compact Symmetric Spaces
本文给出了紧Gelfand对$(G, K)$上的有限$K$ -双不变测度具有平方可积密度的充要条件。对于卷积半群，这等价于在正时间内具有连续密度。当$(G,K)$是紧致的黎曼对称对时，我们研究了对称空间$X = G/K$上$G$不变Feller过程的诱导跃迁密度.这些是作为$K$ -双不变Lévy过程在$G$上的投影得到的，其定律形成卷积半群。我们得到一个傅里叶级数展开的密度，在球函数，其中的频谱是由Gangolli的Lévy-Khintchine公式描述。$X$上任意给定点的返回密度由转移半群的迹给出，对于从属的布朗运动，我们可以利用Bañuelos和Baudoin最近的工作计算这个量的短时渐近性.在球的情况下，与Funk-Hecke定理有一个有趣的联系。
We study perturbations of Feller generators under `lower order terms' with measurable coefficients. We investigate which properties of the original semigroup -- such as positivity, conservativeness and the Feller property -- are passed to the perturbed semigroup. We give several examples and discuss applications in the theory of martingale problems and stochastic differential equations with measurable coefficients.
In this paper, we consider the (upper) semigroup envelope, i.e. the least upper bound, of a given family of linear Feller semigroups. We explicitly construct the semigroup envelope and show that, under suitable assumptions, it yields viscosity solutions to abstract Hamilton-Jacobi-Bellman-type partial differential equations related to stochastic optimal control problems arising in the field of Robust Finance. We further derive conditions for the existence of a Markov process under a nonlinear expectation related to the semigroup envelope for the case where the state space is locally compact. T…
Chernoff approximation of subordinate semigroups and applications
本文利用Beschoff定理来逼近由从属过程构造的发展半群。所考虑的半群从属于一些原始的，未知的明确，但已经近似由相同的方法，相对于从属者的对应物或者已知的转移概率，或已知的和有界的Lévy测度。这些结果被应用于获得相应的半群的Feller过程的从属，（Feller型）扩散在欧几里得空间，星图和黎曼流形的近似。得到的近似是基于明确给出的运营商，因此可以用于直接计算和计算机建模。在几种情况下，所得到的近似作为初等函数的迭代积分给出，并导致所考虑的半群的费曼公式的表示。
We show how Hölder estimates for Feller semigroups can be used to obtain regularity results for solutions to the Poisson equation $Af=g$ associated with the (extended) infinitesimal generator $A$ of a Feller process. The regularity of $f$ is described in terms of Hölder-Zygmund spaces of variable order and, moreover, we establish Schauder estimates. Since Hölder estimates for Feller semigroups have been intensively studied in the last years, our results apply to a wide class of Feller processes, e.g. random time changes of Lévy processes and solutions to Lévy-driven stochastic differential equ…
Markov selections and Feller properties of nonlinear diffusions
本文研究了具有不确定局部特征的非线性扩散，建立了其 Feller 性质，考察了次线性 Markov 半群的线性化与平滑效应，并将其值函数与非线性 Kolmogorov 方程及 Nisio 半群建立联系。
Lagrangian and Hamiltonian Feynman formulae for some Feller semigroups and their perturbations
费曼公式是一个发展方程的初边值问题的解的表示（或者等价地，一个解决问题的半群的表示），它是由一些初等函数的$n$ -折叠迭代积分的极限表示为$n\to\infty$。本文给出了一类与Feller过程相关的半群的Feynman公式。有限维积分的Feynman公式给出近似的功能积分在一些Feynman-Kac公式对应的基本过程。因此，这些费曼公式给出了一个有效的工具来计算函数积分相对于这些费勒过程产生的概率测度，特别是，以获得模拟的费勒过程。
Upper estimates of transition densities for stable-dominated semigroups
对于跳跃强度小于旋转不变稳定Levy过程的Feller半群，我们得到了转移密度的上估计
Chernoff approximations of Feller semigroups and the associated diffusion processes in Riemannian manifolds are studied. The manifolds are assumed to be of bounded geometry, thus including all compact manifolds and also a wide range of non-compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a (generally non-compact) manifold of bounded geometry. A construction of Chernoff approximations is presented for these Feller semigroups in terms of shift operators. This provides approximations of solutions to initial v…
In $\mathbb R^d$, $d \geq 3$, consider the divergence and the non-divergence form operators \begin{equation} \tag{$i$} - \nabla \cdot a \cdot \nabla + b \cdot \nabla, \end{equation} \begin{equation} \tag{$ii$} - a \cdot \nabla^2 + b \cdot \nabla, \end{equation} where $a=I+c \mathsf{f} \otimes \mathsf{f}$, the vector fields $\nabla_i \mathsf{f}$ ($i=1,2,\dots,d$) and $b$ are form-bounded (this includes the sub-critical class $[L^d + L^\infty]^d$ as well as vector fields having critical-order singularities). We characterize quantitative dependence on $c$ and the values of the form-bounds of the …
Pseudo Differential Operators and Markov Semigroups on Compact Lie Groups
我们将紧李群上的伪微分算子的Ruzhansky-Turunen理论扩展为一种工具，它可以用来研究群值马尔可夫过程，其精神是N.Jacob及其合作者在欧几里得空间中的工作。将Feller半群及其生成元和预解式表示为伪微分算子，并将构成Feller半群的算子符号表示为转移核的傅里叶变换.符号显式计算的一些例子，包括费勒过程相关的随机流所产生的解决方案的随机微分方程上的组驱动的Lévy过程。研究了一般李群上的一类Lévy型线性算子，当群是紧的时，它们是伪微分算子，并找到了它们产生对称Dirichlet型的条件.
Strong Feller properties for degenerate SDEs with jumps
在完全Hörmander条件下，证明了由加法从属布朗运动驱动的半群的强Feller性质，其中漂移允许任意增长.为此，我们推广了Malicet-Poly \cite{Ma-Po}和Bally-Caramellino \cite{Ba-Ca}关于Wiener泛函在全变差中的收敛性的一个判据.最后，以一串耦合振子为例进行了验证。
We provide a rather explicit formula for the resolvent of a~concatenation of $N$ processes in terms of their exit laws and certain probability measures characterizing the way the processes are concatenated. As an application, we prove an averaging principle saying that by concatenating asymptotically splittable processes one can approximate Markov chains.
Some Theorems on Feller Processes: Transience, Local Times and Ultracontractivity
本文给出了Feller过程的瞬时性、局部时存在性以及相应的Feller半群的超压缩性的充分条件，这些条件对于Lévy过程是尖锐的.该证明使用了局部对称化技术和Feller过程的特征函数的统一上界。作为一个副产品，我们得到稳定的过程（在R的意义下）。Bass）在具有光滑变指标$α(x)\in(0,2)$的$\R^d$上证明了一个关于指数$α(x)$的瞬变判据;如果$d=1$和$\inf_{x\in\R} α(x)\in (1,2)$，则类稳定过程有局部时.
Exotic B-series representation of the Feller semigroup for Itô diffusions and the MSR path integral
本文研究了一维伊藤扩散费勒半群的时间级数展开，通过异类树的扩展，推导出组合因子的显式表达，建立异类Butcher级数表示，并展示其与MSR路径积分形式的等价性。
This work focuses on a class of stochastic damping Hamiltonian systems with state-dependent switching, where the switching process has a countably infinite state space. After establishing the existence and uniqueness of a global weak solution via the martingale approach under very mild conditions, the paper next proves the strong Feller property for regime-switching stochastic damping Hamiltonian systems by the killing technique together with some resolvent and transition probability identities. The commonly used continuity assumption for the switching rates $q_{kl}(\cdot)$ in the literature i…
A class of multidimensional nonlinear diffusions with the Feller property
本文研究一类具有不确定漂移和确定波动率的多维非线性扩散，建立了其Feller性质，并揭示了平滑效应及其与半线性Kolmogorov方程的联系。
Markov Selection and $W$-strong Feller for 3D Stochastic Primitive Equations
研究了三维随机原始方程在周期边界条件下弱解的一些解析性质。与此模型相关联的鞅问题具有满足马尔可夫性的解族，这是通过一个抽象的选择原则来实现的。马尔可夫性是将转移半群的正则性从小时间推广到任意时间的关键。因此，在一个规则的加性噪声，每个马尔可夫解被证明有一个属性的连续依赖于初始条件，这是从采用弱-强唯一性原理和Bismut-Elworthy-Li公式。
On the strong Feller property of the heat equation on quantum graphs with Kirchoff noise
研究量子图上热方程在基尔霍夫噪声扰动下的强Feller性质，通过树状结构和等边星形图的实例分析，探讨噪声对过渡半群性质的影响以及解的正则性。
Approximation of Stable-dominated Semigroups
我们考虑由旋转不变稳定Lévy测度控制的跳跃系统所决定的算子Feller半群。使用近似模式，我们证明了存在性，并获得相应的热核的估计。
Continuous Branching Processes with Settlement in Cancer Metastasis: Stochastic Modelling and the Feller Property
本文提出了一种记录肿瘤细胞位置的分支过程模型，通过严格构建和马尔可夫性质的建立，研究了Feller性质并推导了显式生成元，从而得到费曼-凯茨型公式。
We consider the stochastic differential equation $dX_t = A(X_{t-}) \, dZ_t$, $ X_0 = x$, driven by cylindrical $α$-stable process $Z_t$ in $R^d$, where $α\in (0,1)$ and $d \ge 2$. We assume that the determinant of $A(x) = (a_{ij}(x))$ is bounded away from zero, and $a_{ij}(x)$ are bounded and Lipschitz continuous. We show that for any fixed $γ\in (0,α)$ the semigroup $P_t$ of the process $X_t$ satisfies $|P_t f(x) - P_t f(y)| \le c t^{-γ/α} |x - y|^γ ||f||_\infty$ for arbitrary bounded Borel function $f$. Our approach is based on Levi's method.
Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker-Planck-Kolmogorov equations
本文研究了由离开给定领域时被杀死的Feller过程生成的半群，这些半群对应于一类演化方程的Cauchy-Dirichlet型初始-外部值问题。通过Chernoff定理对这些半群进行近似，并将构造的Chernoff近似转化为Feynman公式，即n重积分的极限。此外，提出了一种近似时间分数阶（包括分布阶时间分数阶）演化方程解的方法，该方法基于时间分数阶与时间非分数阶演化方程之间的联系以及Chernoff近似。进一步，该方法导致时间分数阶演化方程解的Feynman公式。通过考虑一类分布阶时间分数阶扩散方程，获得了对应的Cauchy和Cauchy-Dirichlet问题的Feynman公式。
Quasi-ergodicity of compact strong Feller semigroups on $L^2$
本文研究无限测度空间上非自伴紧强 Feller 半群的拟遍历性，通过热含量有限性与渐进一致基态支配刻画拟平稳测度唯一性，并应用于非局部 Schrödinger 算子。
On smoothing properties of transition semigroups associated to a class of SDEs with jumps
证明了一类由加性纯跳Lévy噪声驱动的随机微分方程的非局部转移半群的光滑性质.特别地，我们假设驱动解的Lévy过程是从属Wiener过程和独立的任意Lévy过程的和，漂移系数是连续的（但不一定是Lipschitz连续的）并且增长不快于多项式，并且解允许Feller弱解。利用概率和分析相结合的方法，给出了与该算子相关联的马氏半群是强Feller的充分条件以及$L_p$映射到连续有界函数的充分条件.一个关键的中间步骤是研究与从属维纳过程相关的过渡半群在从属子的负矩方面的正则化性质。
Time inhomogeneous Generalized Mehler Semigroups
对实可分Hilbert空间${\mathds{H}}$上的有界可测函数$f$，通过$$ p_{s,t}f(x)=\int_{\mathds{H}} f(U(t,s)x+y)\,μ_{t,s}(dy), \quad t\geq s, \ x\in{\mathds{H}} $$定义了${\mathds{H}}$上的时间非齐次广义Mehler半群，其中$(U(t,s))_{t\geq s}$是${\mathds{H}}$上有界算子的演化族，$(μ_{t,s})_{t\geq s}$是$({\mathds{H}}, \B({\mathds{H}}))$上满足时间非齐次斜卷积方程$$μ_{t,s}=μ_{t,r}*(μ_{r,s}\circ U(t,r)^{-1}),\quad t\geq r\geq s.$$的概率测度族。非自治（可能是非连续）的Ornstein-Uhlenbeck过程的驱动下，一些适当的添加剂过程。给出了$(μ_{t,s})_{t\geq s}$的弱连续性、无穷可除性、相关的“加法过程”、Lévy-Khintchine型表示、构造和谱表示.研究了$(p_{s,t})_{t\geq s}$的测度（=时空不变测度）的演化系统的结构、存在性和唯一性.我们还建立了$(p_{s,t})_{t\geq s}$在Wang（1997，PTRF）意义下的无量纲Harnack不等式.作为Ha…
Genealogy-valued Feller diffusion
本文将 Feller 分支扩散的谱系表示为 $\mathbb{U}$ 值过程，证明鞅问题的适定性与 Feller 性质，并利用拼接、对偶和 Lévy-Khintchine 表示刻画其条件存活及长时间极限。
The e-property of asymptotically stable Markov semigroups
本文研究了 Markov 半群的渐近稳定性与 e-性质的关系，证明了具有非空支撑内部的渐近稳定 Markov-Feller 半群满足最终 e-性质，且强随机连续性下最终 e-性质蕴含 e-性质。
We consider Markov processes that alternate continuous motions and jumps in a general locally compact polish space. Starting from a mechanistic construction, a first contribution of this article is to provide conditions on the dynamics so that the associated transition kernel forms a Feller semigroup, and to deduce the corresponding infinitesimal generator. In a second contribution, we investigate the ergodic properties in the special case where the jumps consist of births and deaths, a situation observed in several applications including epidemiology, ecology and microbiology. Based on a coup…
The strong Feller property for singular stochastic PDEs
证明了一类奇异随机偏微分方程生成的马尔可夫半群满足强Feller性质。这些包括例如KPZ方程和动态$Φ^4_3$模型。作为推论，我们证明了布朗桥测度是具有周期边界条件的KPZ方程的唯一不变测度。
We prove that the operator $L_0=-(1+|x|)^β(-Δ)^{α/2}$ with $α\in(0,2)$, $d>α$ and $β\ge0$ generates a compact semigroup or resolvent on $L^2(\R^d;(1+|x|)^{-β}\,dx)$, if and only if $β>α$. When $β>α$, we obtain two-sided asymptotic estimates for high order eigenvalues, and sharp bounds for the corresponding heat kernel.
The Ornstein Uhlenbeck Bridge and Applications to Markov Semigroups
对于任意希尔伯特空间值的Ornstein-Uhlenbeck过程，我们构造了连接起点$x$和终点$y$的Ornstein-Uhlenbeck桥，该终点属于满测度的某个线性子空间.我们还导出了一个随机发展方程满足的桥，并研究了它的基本性质。然后利用该方法研究了带加性噪声的非线性随机发展方程的马氏转移半群。给出了跃迁密度的显式表达式，并研究了其规律性。在强Feller性质和不变测度存在的条件下，证明了转移半群将$L^p$函数映射为连续函数.我们还证明了对于某些$q>p>1$，特别是Hilbert-Schmidt型的，转移算子是$q$ -求和的。
Law of the Iterated Logarithm for Markov Semigroups with Exponential Mixing in the Wasserstein Distance
本文研究了具有指数混合性的马尔可夫半群在瓦瑟斯坦距离下的迭代对数定律，并探讨了其在无限维随机微分方程中的应用。
We prove that the semigroup generated by the open KPZ equation on a bounded spatial interval with Neumann boundary conditions parametrized by real parameters u and v enjoys the strong Feller property. From this we conclude that for u+v>0, min(u,v)>-1 the stationary measure constructed in Corwin and Knizel (arXiv:2103.12253) is the unique stationary measure for the equation. It is expected that the same conclusion holds for all values of u and v.
Feller generators with singular drifts in the critical range
本文研究了具有临界阶奇异漂移的扩散算子，通过在足够大的 $L^p$ 空间中应用 De Giorgi 方法，在更宽的奇性幅度范围内构造了强连续 Feller 半群，并在临界边界值下于特定的 Orlicz 空间中构造了强连续半群。
Subordinated discrete semigroups of operators
给定巴拿赫空间中的一个幂有界线性算子T和非负整数上的概率F，可以构成一个“从属”算子S = \sum_k F（k）T^k.得到了从属离散半群（S^n：n= 1，2，.）在一定条件下，F.特别是，我们研究概率F的性质，S满足Ritt预解条件时，T是幂有界的。这个属性的例子和反例进行了讨论。T的幂有界性假设有时可以用较弱的Kreiss预解式条件来代替。
Nonlinear Lévy and nonlinear Feller processes: an analytic introduction
V.N.提出了研究一般非线性马尔可夫过程的程序。Kolokoltsov“Nonlinear马尔可夫半群and Interacting Lévy Type Processes”（2004）. Stat. Physics 126：3（2007），585-642），并且由作者在专著“Nonlinear Markov processes and kinetic equations”中开发。剑桥大学出版社，2010年，其中特别介绍了非线性Lévy过程。本文是对非线性马尔可夫过程这一迅速发展的课题的邀请。我们提供了一个快速（同时更抽象）介绍的基本分析方面的理论发展的第二部分，上述书。
Gradient Estimates and Applications for SDEs in Hilbert Space with Multiplicative Noise and Dini Continuous Drift
考虑可分希尔伯特空间中的随机发展方程，该方程具有良好的乘性噪声和局部Dini连续漂移。我们证明了，对于任何初始数据的方程有一个唯一的（可能是爆炸性的）温和的解决方案。在保证解不爆炸的合理条件下，证明了相应的马尔可夫半群的强Feller性质。在一定的整体条件下，得到了相应半群的梯度估计和log-Harnack不等式，这些结果在有限维中也是新的。
Resolvent approach to diffusions with discontinuous scale
本文研究尺度函数s仅非降时算子L=1/2 D_m D_s对应的马氏过程，证明其再生核由α-调和方程的两个正单调解生成，并诱导出与半群或Dirichlet型方法相同的马氏过程。
Exact moduli of continuity for the local times of Feller Brownian motions
该研究延续前期工作，利用Eisenbaum同构定理，得到半直线上Feller布朗运动在空间区间$(0,1]$局部时的精确一致连续模，并推广至$[0,1]$及更一般扩散过程情形。
Motivated by models of signaling pathways in B lymphocytes, which have extremely large nuclei, we study the question of how reaction-diffusion equations in thin $2D$ domains may be approximated by diffusion equations in regions of smaller dimensions. In particular, we study how transmission conditions featuring in the approximating equations become integral parts of the limit master equation. We device a scheme which, by appropriate rescaling of coefficients and finding a common reference space for all Feller semigroups involved, allows deriving the form of the limit equation formally. The res…
We use analytical methods to construct the two-parameter Feller semigroup associated with a Markov process on a line with a moving membrane such that at the points on both sides of the membrane it coincides with the ordinary diffusion processes given there and its behavior after reaching the membrane is described by a kind of nonlocal Feller-Wentzell conjugation condition.
On some results on the stability of Markov operators
给出了度量空间上定义的具有有界连续函数的e-性质的Feller半群的不变测度存在性的一个判据，并利用它证明了满足下界条件的半群的渐近稳定性.我们的结果补充了A. Lasota和J.A. Yorke和T.波兰空间中的沙雷克。
Asymptotic stability for a class of Markov semigroups
设$U\subset K$是紧致度量空间的开稠密子集，$\{Φ_t\}_{t\ge0}$是$U$上的有界Borel可测函数空间上的Markov半群，具有强Feller性质.假设对于每个$x\in\bdu$，在$x$处存在屏障$h\in C(K)$，使得对于所有$t\ge0$都存在$Φ_t(h)\ge h$。还假设每个实值$g\in C(K)$，对于所有$t\ge0$具有$Φ_t(g)\ge g$，并且在$U$内的一个点处达到其全局最大值是常数。则对于每个$f\in C(K)$，存在一致极限$F=\lim_{t\to\infty}Φ_t(f)$。此外，$F$在$K$上是连续的，与$f$在$\partial{U}$上和$Φ_t(F)=F$在所有$t\ge0$上一致。
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