Feller Processes on non-locally compact spaces

Doubly Feller property of Brownian motions with Robin boundary condition

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.

Construction of $\mathcal L^p$-strong Feller Processes via Dirichlet Forms and Applications to Elliptic Diffusions

We provide a general construction scheme for $\mathcal L^p$-strong Feller processes on locally compact separable metric spaces. Starting from a regular Dirichlet form and specified regularity assumptions, we construct an associated semigroup and resolvents of kernels having the $\mathcal L^p$-strong Feller property. They allow us to construct a process which solves the corresponding martingale problem for all starting points from a known set, namely the set where the regularity assumptions hold. We apply this result to construct elliptic diffusions having locally Lipschitz matrix coefficients …

非齐次Hawkes过程的收敛性与Feller随机测度

Convergence of nonhomogeneous Hawkes processes and Feller random measures

研究非齐次Hawkes过程在近不稳定极限区域的缩放极限，其激发测度可能依赖于世代，通过非线性卷积方程刻画极限随机测度，扩展了Feller扩散和分数Feller（CIR）过程，并允许一般Levy型算子的驱动噪声。

关于非紧对称空间上Fréchet均值的Kolmogorov-Feller弱大数定律

On the Kolmogorov-Feller weak law of large numbers for the Fréchet mean on non-compact symmetric spaces

针对非紧对称空间上的样本Fréchet均值，证明了Kolmogorov-Feller弱大数定律，适用于独立非同分布数据，并给出对称正定矩阵和乘积对称空间的例子。

Lévy-type processes: convergence and discrete schemes

We characterise the convergence of a certain class of discrete time Markov processes toward locally Feller processes in terms of convergence of associated operators. The theory of locally Feller processes is applied to Lévy-type processes in order to obtain convergence results on discrete and continuous time indexed processes, simulation methods and Euler schemes. We also apply the same theory to a slightly different situation, in order to get results of convergence of diffusions or random walks toward singular diffusions. As a consequence we deduce the convergence of random walks in random me…

On the entrance at infinity of Feller processes with no negative jumps

Consider a non-explosive positive Feller process with no negative jumps. It is shown in this note that when infinity is an entrance boundary, in the sense that the entrance times of the process remain bounded when the initial value tends to infinity, the process admits a Feller extension on the compactified state space $[0,\infty]$. Moreover, when started from infinity, the extended Markov process on $[0,\infty]$ leaves infinity instantaneously and stays finite, almost-surely. Arguments are adapted from a proof given by O. Kallenberg for diffusions. We also show that the process started from $…

关于 Feller Markov 链的不变分布及其在随机切换动力系统中的应用

On invariant distributions of Feller Markov chains with applications to dynamical systems with random switching

本文给出保证紧流形上 Feller Markov 链不变分布具有不同正则性密度的条件，并将其用于随机复合映射与随机切换流形成的过程。

Feller and ergodic properties of jump-move processes with applications to interacting particle systems

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…

Solutions of Lévy-driven SDEs with unbounded coefficients as Feller processes

Let $(L_t)_{t \geq 0}$ be a $k$-dimensional Lévy process and $σ: \mathbb{R}^d \to \mathbb{R}^{d \times k}$ a continuous function such that the Lévy-driven stochastic differential equation (SDE) $$dX_t = σ(X_{t-}) \, dL_t, \qquad X_0 \sim μ$$ has a unique weak solution. We show that the solution is a Feller process whose domain of the generator contains the smooth functions with compact support if, and only if, the Lévy measure $ν$ of the driving Lévy process $(L_t)_{t \geq 0}$ satisfies $$ν(\{y \in \mathbb{R}^k; |σ(x)y+x|<r\}) \xrightarrow[]{|x| \to \infty} 0.$$

An Introduction to Lévy and Feller Processes. Advanced Courses in Mathematics - CRM Barcelona 2014

These lecture notes are an extended version of my lectures on Lévy and Lévy-type (Feller) processes given at the "Second Barcelona Summer School on Stochastic Analysis" 2014 organized by the Centre de Recerca Matemaatica (CRM). The lectures are aimed at advanced graduate and PhD students. In order to read these notes, one should have sound knowledge of measure theoretic probability theory and some background in stochastic processes, as it is covered in my books "Measures, Integals and Martingales" (Cambridge University Press) and "Brownian Motion" (de Gruyter). My purpose in these lectures is …

Existence, uniqueness and ergodic properties for time-homogeneous Itô-SDEs with locally integrable drifts and Sobolev diffusion coefficients

Using elliptic and parabolic regularity results in $L^p$-spaces and generalized Dirichlet form theory, we construct for every starting point weak solutions to SDEs in $\mathbb{R}^d$ up to their explosion times including the following conditions. For arbitrary but fixed $p>d$ the diffusion coefficient $A=(a_{ij})_{1\le i,j\le d}$ is locally uniformly strictly elliptic with functions $a_{ij}\in H^{1,p}_{loc}(\mathbb{R}^d)$ and the drift coefficient $\mathbf{G}=(g_1,\dots, g_d)$ consists of functions $g_i\in L^p_{loc}(\mathbb{R}^d)$. The solution originates by construction from a Hunt process wit…

广义Feller理论的衍生结果

Ramifications of generalized Feller theory

本文扩展了广义Feller过程的构造与路径性质等民间结果并给出完整证明，引入扩展Feller过程新概念，通过特征线法将其与加权空间上的传输方程和连续半流联系，并用反例证明基本定义各条件均不可省略。

从泛函分析视角看超Feller算子

Ultra Feller operators from a functional analytic perspective

本研究从泛函分析角度证明Polish空间上两强Feller算子乘积的超Feller性质，去掉了算子正性假设并拓展适用状态空间，指出其是Dunford--Pettis性质Banach空间上弱紧算子乘积为紧算子定理的变体。

Pseudo Differential Operators and Markov Semigroups on Compact Lie Groups

We extend the Ruzhansky-Turunen theory of pseudo differential operators on compact Lie groups into a tool that can be used to investigate group-valued Markov processes in the spirit of the work in Euclidean spaces of N.Jacob and collaborators. Feller semigroups, their generators and resolvents are exhibited as pseudo-differential operators and the symbols of the operators forming the semigroup are expressed in terms of the Fourier transform of the transition kernel. The symbols are explicitly computed for some examples including the Feller processes associated to stochastic flows arising from …

Chernoff approximations of Feller semigroups in Riemannian manifolds

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…

Spectral Asymptotics for Krein-Feller-Operators with respect to Random Recursive Cantor Measures

We study the limit behavior of the Dirichlet and Neumann eigenvalue counting function of generalized second order differential operators $\frac{d}{d μ} \frac{d}{d x}$, where $μ$ is a finite atomless Borel measure on some compact interval $[a,b]$. We firstly recall the results of the spectral asymptotics for these operators received so far. Afterwards, we give the spectral asymptotics for so called random recursive Cantor measures. Finally, we compare the results for random recursive and random homogeneous Cantor measures.

Brownian Motions on Metric Graphs with Non-Local Boundary Conditions II: Construction

A pathwise construction of discontinuous Brownian motions on metric graphs is given for every possible set of non-local Feller-Wentzell boundary conditions. This construction is achieved by locally decomposing the metric graphs into star graphs, establishing local solutions on these partial graphs, pasting the solutions together, introducing non-local jumps, and verifying the generator of the resulting process.

$L^2$ 上紧强 Feller 半群的拟遍历性

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。

通过测度空间变换的广义Krein-Feller算子与间隙扩散

Generalised Krein-Feller operators and gap diffusions via transformations of measure spaces

该研究通过测度分布函数构造的等距同构，将广义Krein-Feller算子的特征值问题转化为经典情形，得到上谱维数新刻画，整合谱渐近结果并复原相关广义间隙扩散的性质与联系。

On the nonexplosion and explosion for nonhomogeneous Markov pure jump processes

In this paper, we obtain new drift-type conditions for nonexplosion and explosion for nonhomogeneous Markov pure jump processes in Borel state spaces. The conditions are sharp; e.g., the one for nonexplosion is necessary if the state space is in addition locally compact and the $Q$-function satisfies weak Feller-type and local boundedness conditions. We comment on the relations of our conditions with the existing ones in the literature, and demonstrate some possible applications.

Approximation in law of locally $α$-stable Lévy-type processes by non-linear regressions

We study a real-valued Lévy-type process $X$, which is locally $α$-stable in the sense that its jump kernel is a combination of a `principal' (state dependent) $α$-stable part with a `residual' lower order part. We show that under mild conditions on the local characteristics of a process (the jump kernel and the velocity field) the process is uniquely defined, is Markov, and has the strong Feller property. We approximate $X$ in law by a non-linear regression $\widetilde X^x_{t}=\mathfrak{f}_t(x)+t^{1/α}U^{x}_t$ with a deterministic regressor term $\mathfrak{f}_t(x)$ and $α$-stable innovation t…

具有局部弱单调性系数的McKean-Vlasov随机偏微分方程：存在性、唯一性、遍历性、指数混合与极限定理

McKean-Vlasov SPDEs with coefficients exhibiting locally weak monotonicity: existence, uniqueness, ergodicity, exponential mixing and limit theorems

本文针对具有局部弱单调性系数的McKean-Vlasov SPDE，利用Galerkin投影和广义耦合方法证明了解的存在唯一性，进而建立指数遍历性、指数混合性以及强大数定律和中心极限定理。

Gradient Estimates and Applications for SDEs in Hilbert Space with Multiplicative Noise and Dini Continuous Drift

Consider the stochastic evolution equation in a separable Hilbert space with a nice multiplicative noise and a locally Dini continuous drift. We prove that for any initial data the equation has a unique (possibly explosive) mild solution. Under a reasonable condition ensuring the non-explosion of the solution, the strong Feller property of the associated Markov semigroup is proved. Gradient estimates and log-Harnack inequalities are derived for the associated semigroup under certain global conditions, which are new even in finite-dimensions.

在区域边界处被杀死的退化过程

Degenerate processes killed at the boundary of a domain

本文研究退出相对紧集时被杀死的退化Feller过程，给出其存在拟平稳分布的一般条件及唯一性、指数收敛条件，并将结果应用于非椭圆和亚椭圆随机微分方程。

开放KPZ方程的遍历性结果

Ergodicity results for the open KPZ equation

本文为开放边界KPZ方程的不变测度提供了存在性新证明和唯一性双证明，结合强Feller性质得到全变差指数收敛，并构造紧致状态空间以扩展适用范围。

关于局部单调随机演化方程的遍历性与混合性

Ergodicity and mixing for locally monotone stochastic evolution equations

本文研究了局部单调驱动和退化加性Wiener噪声的随机演化方程，建立了存在唯一不变概率测度的定量条件，并提供了解的矩估计和半群的e-性质，以及2-Wassersteinε-混合时间的上界。

$L^2$ Properties of Lévy Generators on Compact Riemannian Manifolds

We consider isotropic Lévy processes on a compact Riemannian manifold, obtained from an $\mathbb{R}^d$-valued Lévy process through rolling without slipping. We prove that the Feller semigroups associated with these processes extend to strongly continuous contraction semigroups on $L^p$, for $1\leq p<\infty$, and that they are self-adjoint when $p=2$. When the motion has a non-trivial Brownian part, we prove that the generator has a discrete spectrum of eigenvalues and that the semigroup is trace-class.

Canonical diffusions on the pattern spaces of aperiodic Delone sets

We consider pattern spaces of aperiodic and repetitive Delone sets of finite local complexity. These spaces are compact metric spaces and constitute a special class of foliated spaces. We define new Sobolev spaces with respect to the unique invariant measure and prove the existence of the unitary Schrödinger semigroup, which in physics terms describe the evolution of phasons. We define and study natural leafwise diffusion processes on these pattern spaces. These processes have Feller, but lack strong Feller and hypercontractivity properties, and heat kernels do not exist. The associated Dirich…

Krein-Feller型算子的谱理论及其在随机分数椭圆和抛物方程中的应用

Spectral Theory of Krein-Feller Type Operators and Applications in Stochastic Fractional Elliptic and Parabolic Equations

本文研究Krein-Feller型算子的谱理论，推导其特征值的渐近下界和收敛指数上界，并应用于随机分数椭圆和抛物方程的存在性问题。

A note on Gromov-Hausdorff-Prokhorov distance between (locally) compact measure spaces

We present an extension of the Gromov-Hausdorff metric on the set of compact metric spaces: the Gromov-Hausdorff-Prokhorov metric on the set of compact metric spaces endowed with a finite measure. We then extend it to the non-compact case by describing a metric on the set of rooted complete locally compact length spaces endowed with a locally finite measure. We prove that this space with the extended Gromov-Hausdorff-Prokhorov metric is a Polish space. This generalization is needed to define Lévy trees, which are (possibly unbounded) random real trees endowed with a locally finite measure.

Non-autonomous interacting particle systems in continuum

A conservative Feller evolution on continuous bounded functions is constructed from a weakly continuous, time-inhomogeneous transition function describing a pure jump process on a locally compact Polish space. The transition function is assumed to satisfy a Foster-Lyapunov type condition. The results are applied to interacting particle systems in continuum, in particular to general birth-and-death processes (including jumps). Particular examples such as the BDLP and Dieckmann-Law model are considered in the end.

Idempotent states on locally compact groups and quantum groups

This is a short survey on idempotent states on locally compact groups and locally compact quantum groups. The central topic is the relationship between idempotent states, subgroups and invariant C*-subalgebras. We concentrate on recent results on locally compact quantum groups, but begin with the classical notion of idempotent probability measure. We also consider the `intermediate' case of idempotent states in the Fourier--Stieltjes algebra: this is the dual case of idempotent probability measures and so an instance of idempotent states on a locally compact quantum group.

局部平衡马尔可夫过程的基础

Foundations of locally-balanced Markov processes

本文引入并研究一般状态空间上的局部平衡马尔可夫跳跃过程（LBMJPs），解决离散参数及非光滑分布采样局限，确立其适定性、遍历性等性质，推导指数遍历条件与扩散极限，拓展蒙特卡洛采样应用。

Harmonic functions on locally compact groups of polynomial growth

We extend a theorem by Kleiner, stating that on a group with polynomial growth, the space of harmonic functions of polynomial of at most $k$ is finite dimensional, to the settings of locally compact groups equipped with measures with non-compact support. This has implications to the structure of the space of polynomially growing harmonic functions.