Vassili Kolokoltsov
The theory of monotonicity and duality is developed for general one-dimensional Feller processes. Moreover it is shown that local monotonicity conditions (conditions on the Lévy kernel) are sufficient to prove the well-posedness of the corresponding Markov semigroup and process, including unbounded coefficients and processes on the half-line.
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We analyze the notions of monotonicity and complete monotonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide in continuous time but not in discrete-time.
We introduce a notion of $k$th order stochastic monotonicity and duality that allows one to unify the notion used in insurance mathematics (sometimes refereed to as Siegmund's duality) for the study of ruin probability and the duality responsible for the so-called put - call symmetries in option pricing. Our general $k$th order duality can be financially interpreted as put - call symmetry for powered options. The main objective of the present paper is to develop an effective analytic approach to the analysis of duality leading to the full characterization of $k$th order duality of Markov proce…
This note presents a simple proof of the monotonicity of the invariant distribution of a discrete Markov chain with a finite state space. This answers a question recently raised by David Siegmund.
We formalize and analyze the notions of stochastic monotonicity and realizable mono-tonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which stochastic monotonicity and realizable monotonicity coincide in continuous-time but not in discrete-time.
A complete characterization of monotonicity equivalence for continuous-time Markov processes
本文证明了Dai Pra等人关于有限偏序集上连续时间马尔可夫过程单调性等价的猜想，并给出了单调性等价偏序集的完整刻画。
In this paper we show that we can use Markov kernels as a model for optimal transport. This new framework can be easily translated into the standard coupling formulation of optimal transport, and we show that we can use a coupling as a Markov kernel for many optimal transport problems. Using kernels allows us to extend optimal transport to signed measures and treats the support of the measure as the salient feature. This approach reveals additional structure for one-dimensional signed optimal transport.
We construct strong stationary dual chains for Ising model on a circle, non-symmetric random walk on square lattice and a random walk on hypercube. The strong stationary dual chains are all sharp and have the same state space as original chains. We use Möbius monotonicity of these chains with respect to natural orderings of the corresponding state spaces. This method provides an easy way to find eigenvalues in the Ising model and for a random walk on hypercube. Keywords: Markov chains; stochastic monotonicity; eigenvalues; Möbius monotonicity; strong stationary duality; strong stationary times…
For Markov chains with a partially ordered finite state space we show strong stationary duality under the condition of Möbius monotonicity of the chain. We show relations of Möbius monotonicity to other definitions of monotone chains. We give examples of dual chains in this context which have transitions only upwards. We illustrate general theory by an analysis of nonsymmetric random walks on the cube with an application to networks of queues.
Multi-class queueing networks (McQNs) extend the classical concept of Jackson network by allowing jobs of different classes to visit the same server. While such a generalization seems rather natural, from a structural perspective there is a significant gap between the two concepts. Nice analytical features of Jackson networks, such as stability conditions, product-form equilibrium distributions, and stochastic monotonicity do not immediately carry over to the multi-class framework. The aim of this paper is to shed some light on this structural gap, focusing on monotonicity properties. To this …
Given two random variables $X$ and $Y$, stochastic monotonicity describes a monotone influence of $X$ on $Y$. We prove two different characterizations of stochastically monotone $2$-copulas using the isomorphism between $2$-copulas and Markov operators. The first approach establishes a one-to-one correspondence between stochastically monotone copulas and monotonicity-preserving Markov operators. The second approach characterizes stochastically monotone copulas by their monotonicity property with respect to the Markov product. Applying the latter result, we identify all idempotent stochasticall…
We study stochastic monotonicity and propagation of order for Markov processes with respect to stochastic integral orders characterized by cones of functions satisfying $Φf \geq 0$ for some linear operator $Φ$. We introduce a new functional analytic technique based on the generator $A$ of a Markov process and its resolvent. We show that the existence of an operator $B$ with positive resolvent such that $ΦA - B Φ$ is a positive operator for a large enough class of functions implies stochastic monotonicity. This establishes a technique for proving stochastic monotonicity and propagation of order…
We provide a nonasymptotic analysis of convergence to stationarity for a collection of Markov chains on multivariate state spaces, from arbitrary starting points, thereby generalizing results in [Khare and Zhou Ann. Appl. Probab. 19 (2009) 737-777]. Our examples include the multi-allele Moran model in population genetics and its variants in community ecology, a generalized Ehrenfest urn model and variants of the Polya urn model. It is shown that all these Markov chains are stochastically monotone with respect to an appropriate partial ordering. Then, using a generalization of the results in [D…
The optimal weak transport problem has recently been introduced by Gozlan et.\ al. We provide general existence and duality results for these problems on arbitrary Polish spaces, as well as a necessary and sufficient optimality criterion in the spirit of cyclical monotonicity. As an application we extend the Brenier-Strassen Theorem of Gozlan-Juillet to general probability measures on $R^d$ under minimal assumptions. A driving idea behind our proofs is to consider the set of transport plans with a new (`adapted') topology which seems better suited for the weak transport problem and allows to c…
Processes having the same bridges are said to belong to the same reciprocal class. In this article we analyze reciprocal classes of Markov counting processes by identifying their reciprocal invariants and we characterize them as the set of counting processes satisfying some duality formula.
For a given absorbing Markov chain $X^*$ on a finite state space, a chain $X$ is a sharp antidual of $X^*$ if the fastest strong stationary time of $X$ is equal, in distribution, to the absorption time of $X^*$. In this paper we show a systematic way of finding such an antidual based on some partial ordering of the state space. We use a theory of strong stationary duality developed recently for Möbius monotone Markov chains. We give several sharp antidual chains for Markov chain corresponding to a generalized coupon collector problem. As a consequence - utilizing known results on a limiting di…
We present a theorem which elucidates the connection between self-duality of Markov processes and representation theory of Lie algebras. In particular, we identify sufficient conditions such that the intertwining function between two representations of a certain Lie algebra is the self-duality function of a (Markov) operator. In concrete terms, the two representations are associated to two operators in interwining relation. The self-dual operator, which arise from an appropriate symmetric linear combination of them, is the generator of a Markov process. The theorem is applied to a series of ex…
This is the first part of a possible monograph on the duality of Markov processes. It contains a proof of Fitzsimmons' existence theorem of a moderate Markov dual process relative to an excessive measure, m, together with the necessary preliminary material. Then this is applied to prove the correspondence between optional copredictable homogenous random measures and sigma finite measures not charging m-exceptional sets again following Fitzsimmons. The second part which may never be written would deal with duality proper including results from, but not limited to, my joint paper with P. J. Fitz…
The paper is devoted to a systematic study of the duality of processes in the sense that $E f(X_t^x,y)=E f (x, Y_t^y)$ for a certain $f$. This classical topic has well known applications in interacting particles, intertwining, superprocesses, stochastic monotonicity, exit - entrance laws, ruin probabilities in finances, etc. Aiming mostly at the case of $f$ depending on the difference of its arguments, we shall give a systematic study of duality via the analysis of the generators of dual Markov processes leading to various results and insights.
We work out some relations between duality and intertwining in the context of discrete Markov chains, fixing up the background of previous relations first established for birth and death chains and their Siegmund duals. In view of the results, the monotone properties resulting from the Siegmund dual of birth and death chains are revisited in some detail, with emphasis on the non neutral Moran model. We also introduce an ultrametric type dual extending the Siegmund kernel. Finally we discuss the sharp dual, following closely the Diaconis-Fill study.
We show that the multi-species higher spin stochastic vertex model, also called the U_q(A_n^{(1)}) vertex model, satisfies a duality where the indicator function has the form {η^x_{[1,n]} \geq ξ^x_{[1,n]} }. In other words, for every particle in the ξconfiguration of species i at vertex x, there must be a particle of species j > i at vertex x in the ηconfiguration. The proof follows by applying charge reversal to previously discovered duality functions, which also results in open boundary conditions. As a corollary, we recover the duality for the stochastic six vertex model recently found by Y…
Markov branching processes with disasters: extinction, survival and duality to p-jump processes
本文通过建立 p-跳跃过程在单位区间上的极限定理，并利用概率生成函数对偶性，研究带二项灾难的分支过程的生存概率极限，进而推广到时间非齐次生灭过程和带 p-跳跃的连续状态分支过程。
One of the key ingredients to successfully apply Stein's method for distributional approximation are solutions to the Stein equations and their derivatives. Using Barbour's generator approach, one can solve for the solutions to the Stein equation in terms of the semi-group of a Markov process, which is typically a diffusion process if it is a continuous distribution. For an arbitrary diffusion it can a difficult task to evaluate the semi-group and its derivatives. In this paper, for polynomial test functions, instead of calculating the semi-group of a diffusion, via a duality argument, we inst…
It was recently proven that the correlation function of the stationary version of a reflected Lévy process is nonnegative, nonincreasing and convex. In another branch of the literature it was established that the mean value of the reflected process starting from zero is nonnegative, nondecreasing and concave. In the present paper it is shown, by putting them in a common framework, that these results extend to substantially more general settings. Indeed, instead of reflected Lévy processes, we consider a class of more general stochastically monotone Markov processes. In this setup we show monot…
We establish the existence (and in an appropriate sense uniqueness) of Markovian solutions for ergodic BSDEs under a novel monotonicity condition. Our monotonicity condition allows us to prove existence even when the driver f has arbitrary (in particular super-quadratic) growth in z, which reveals an interesting trade-off between monotonicity and growth for ergodic BSDEs. The technique of proof is to establish a probabilistic representation of the derivative of the Markovian solution, and then use this representation to obtain a-priori estimates. Our study is motivated by applications to ergod…
Positive Markov processes in Laplace duality
本文建立了一个关于正马尔可夫过程拉普拉斯对偶的一般框架，证明了过程具有拉普拉斯对偶当且仅当满足某种完全单调性条件，并探讨了在边界0和∞处的弱连续性和吸收性性质。
We show that any $\mathbb{R}^d\setminus\{0\}$-valued self-similar Markov process $X$, with index $α>0$ can be represented as a path transformation of some Markov additive process (MAP) $(θ,ξ)$ in $S_{d-1}\times\mathbb{R}$. This result extends the well known Lamperti transformation. Let us denote by $\widehat{X}$ the self-similar Markov process which is obtained from the MAP $(θ,-ξ)$ through this extended Lamperti transformation. Then we prove that $\widehat{X}$ is in weak duality with $X$, with respect to the measure $π(x/\|x\|)\|x\|^{α-d}dx$, if and only if $(θ,ξ)$ is reversible with respect …
We prove that Schütz's ASEP Markov duality functional is also a Markov duality functional for the stochastic six vertex model. We introduce a new method that uses induction on the number of particles to prove the Markov duality.
Quantitative Convergence Rates for Stochastically Monotone Markov Chains
本文针对随机单调马尔可夫链，利用序理论混合条件，给出了分布偏差的定量收敛界，并证明经典全变差界是其特例。
We develop the theory of strong stationary duality for diffusion processes on compact intervals. We analytically derive the generator and boundary behavior of the dual process and recover a central tenet of the classical Markov chain theory in the diffusion setting by linking the separation distance in the primal diffusion to the absorption time in the dual diffusion. We also exhibit our strong stationary dual as the natural limiting process of the strong stationary dual sequence of a well chosen sequence of approximating birth-and-death Markov chains, allowing for simultaneous numerical simul…
Stochastic duality and eigenfunctions
本文基于马尔可夫生成元共享特征值构造对偶函数，通过谱分解完整刻画有限状态空间的对偶关系，并应用于西格蒙德对偶及可解例子，提取谱信息并构造所有对偶函数。
We provide a systematic study of the notion of duality of Markov processes with respect to a function. We discuss the relation of this notion with duality with respect to a measure as studied in Markov process theory and potential theory and give functional analytic results including existence and uniqueness criteria and a comparison of the spectra of dual semi-groups. The analytic framework builds on the notion of dual pairs, convex geometry, and Hilbert spaces. In addition, we formalize the notion of pathwise duality as it appears in population genetics and interacting particle systems. We d…
We show that quantum Schur-Weyl duality leads to Markov duality for a variety of asymmetric interacting particle systems. In particular, we consider three cases: (1) Using a Schur-Weyl duality between a two-parameter quantum group and a two-parameter Hecke algebra from arXiv:math/0108038, we recover the Markov self-duality of multi-species ASEP previously discovered in arXiv:1605.00691 and arXiv:1606.04587. (2) From a Schur-Weyl duality between a co-ideal subalgebra of a quantum group and a Hecke algebra of type B arXiv:1609.01766, we find a Markov duality for a multi-species open ASEP on the …
Downward conditional monotonicity gives survival and extinction for contact processes in random environments
引入马尔可夫调制泊松过程的向下条件单调性，通过标准泊松过程的最优随机占优，推导出接触过程在有限状态随机环境中的生存与灭绝条件。
Mosco convergence of independent particles and applications to particle systems with self-duality
本文证明了独立马尔可夫过程在 Mosco 意义下的 Dirichlet 形式收敛性可推广至多维独立副本，并将该结果应用于具有马尔可夫矩对偶性的交互粒子系统。
Monotonicity of Markov chain transition probabilities via quasi-stationarity -- an application to Bernoulli percolation on $C_k \times Z$
该研究通过马尔可夫链准平稳分布的收敛速率界得到转移概率单调性阈值的显式界，并将其应用于柱图伯努利渗流，证明了连通概率的一致单调性结果。
Duality for Stochastic Control with non-Markovian Random Coefficients
针对含两类随机性且非马尔可夫随机系数仅适配其中一类的随机最优控制问题，开发新对偶方法，通过路径型最优控制问题重获马尔可夫性，得到并求解粗糙二阶HJB方程。
Based on the multidimensional irreducible paving of De March & Touzi, we provide a multi-dimensional version of the quasi sure duality for the martingale optimal transport problem, thus extending the result of Beiglböck, Nutz & Touzi. Similar, we also prove a disintegration result which states a natural decomposition of the martingale optimal transport problem on the irreducible components, with pointwise duality verified on each component. As another contribution, we extend the martingale monotonicity principle to the present multi-dimensional setting. Our results hold in dimensions 1, 2, and…
Monotonicity properties for Bernoulli percolation on layered graphs -- a Markov chain approach
针对分层图上伯努利渗流的连通概率单调性猜想，提出逐层构建渗流构型的马尔可夫链方法，证明有限基图下该单调性对足够大的层差成立，为相关渗流问题提供新研究思路。
Geometry and Duality of Alternating Markov Chains
本文将一般马尔可夫链的半步实现为联合概率分布凸集间关于逆向Kullback-Leibler散度的交替投影，并以此几何刻画证明了奇偶半步定义的马尔可夫链间的信息论对偶性。
Consider a system $X = ((x_ξ(t)), ξ\in Ω_N)_{t \geq 0}$ of interacting Fleming-Viot diffusions with mutation and selection which is a strong Markov process with continuous paths and state space $(\CP(\I))^{Ω_N}$, where $\I$ is the type space, ${Ω_N}$ the geographic space is assumed to be a countable group and $\CP$ denotes the probability measures. We establish various duality relations for this process. These dualities are function-valued processes which are driven by a coalescing-branching random walk, that is, an evolving particle system which in addition exhibits certain changes in the fun…
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。
Interacting particle systems are continuous time Markov processes which are used to construct models in many disciplines. Monotonicity is a property that some interacting particle systems possess. A monotone interacting particle system is called attractive. A benefit of attractiveness is that it simplifies certain calculations, one of which is the ability to use some computational algorithms to sample exactly from the stationary distribution of an ergodic process. Monotonicity is well understood for spin systems such as the contact process. Spin systems only include two particle types however,…
Schur--Weyl duality for diagonalizing a Markov chain on the hypercube
本文利用Schur-Weyl对偶等工具，为超立方体上特定马尔可夫链提供显式正交基，并确定其收敛到平稳分布的精确速率。
On the duality between consensus problems and Markov processes, with application to delay systems
本文通过将多智能体共识问题与马尔可夫过程对偶，提出统一框架处理延迟系统中的时间延迟问题。
Markov duality and Bethe ansatz formula for half-line open ASEP
该研究借助整数线ASEP的马尔可夫对偶推导出半直线及线段开ASEP的对偶性，用贝特拟设求解半直线情形ODE得到电流q-矩积分公式，验证并推导了不同边界下乘性噪声随机热方程的矩公式。
We prove the existence of the persistence exponent $$\logλ:=\lim_{n\to\infty}\frac{1}{n}\log \mathbb{P}_μ(X_0\in S,\ldots,X_n\in S)$$ for a class of time homogeneous Markov chains $\{X_i\}_{i\geq 0}$ taking values in a Polish space, where $S$ is a Borel measurable set and $μ$ is an initial distribution. Focusing on the case of AR($p$) and MA($q$) processes with $p,q\in \mathbb{N}$ and continuous innovation distribution, we study the existence of $λ$ and its continuity in the parameters of the AR and MA processes, respectively, for $S=\mathbb{R}_{\geq 0}$. For AR processes with log-concave inno…
Resolvent intertwining and spectral duality in Markov chains with geometric resetting
该研究揭示带几何重置的马尔可夫链中谱对偶性的预解起源，确定通用临界值$C^*=1/(1+\sqrt{K})$，刻画$(\sigma,\kappa)$可逆链类别，经数值实验验证理论并建立算子理论基础。
Inspired by the recent work of Bertini and Posta, who introduced the boundary driven Brownian gas on $[0,1]$, we study boundary driven systems of independent particles in a general setting, including particles jumping on finite graphs and diffusion processes on bounded domains in $\mathbb{R}^d$. We prove duality with a dual process that is absorbed at the boundaries, thereby creating a general framework that unifies dualities for boundary driven systems in the discrete and continuum setting. We use duality first to show that from any initial condition the systems evolve to the unique invariant…
Markov dualities via the spectral decompositions of the two Markov generators in their bi-orthogonal basis of right and left eigenvectors
本文通过两个马尔可夫生成元在双正交基下的谱分解重新审视马尔可夫对偶性，揭示时间反转与西格蒙德对偶的谱本质，分析赖特-费雪扩散与金曼过程的矩对偶，并展示如何构建新对偶，如非退化实特征值下的定向跳跃过程。
We develop a necessary stochastic maximum principle for a finite-dimensional stochastic control problem in infinite horizon under a polynomial growth and joint monotonicity assumption on the coefficients. The second assumption generalizes the usual one in the sense that it is formulated as a joint condition for the drift and the diffusion term. The main difficulties concern the construction of the first and second order adjoint processes by solving backward equations on an unbounded time interval. The first adjoint process is characterized as a solution to a backward SDE, which is well-posed t…
In 2013, Lu and Ren \cite {luren} considered anticipated backward stochastic differential equations driven by finite state, continuous time Markov chain noise and established the existence and uniqueness of the solutions of these equations and a scalar comparison theorem. In this paper, we provide an estimate for their solutions and study the duality between these equations and stochastic differential delayed equations with Markov chain noise. Finally we derive another comparison theorem for these solutions depending only on the two drivers.
In this paper we derive intertwining relations for a broad class of conservative particle systems both in discrete and continuous setting. Using the language of point process theory, we are able to derive a natural framework in which duality and intertwining can be formulated. We prove falling factorial and orthogonal polynomial intertwining relations in a general setting. These intertwinings unite the previously found classical and orthogonal self-dualities in the context of discrete particle systems and provide new dualities for several interacting systems in the continuum. We also introduce…
We consider a large family of discrete and continuous time controlled Markov processes and study an ergodic risk-sensitive minimization problem. Under a blanket stability assumption, we provide a complete analysis to this problem. In particular, we establish uniqueness of the value function and verification result for optimal stationary Markov controls, in addition to the existence results. We also revisit this problem under a near-monotonicity condition but without any stability hypothesis. Our results also include policy improvement algorithms both in discrete and continuous time frameworks.
For given two standard processes with no positive jumps, we construct, using the excursion theory, a Markov process whose positive and negative motions have the same law as the two processes. The resulting process is a generalization of Kyprianou--Loeffen's refracted Lévy processes. We discuss approximation problem for our refracted processes coming from Lévy processes by removing small jumps and taking the limit as the removal level tends to zero. We also discuss conditions for refracted processes to have dual processes.
Monotone Multiple Stopping and Last-Success Problems
本文提出有限时域离散时间多重停止理论，区分一般递归动态规划与单调结构，构造最优停止时间向量，并应用于多重选择最后成功问题，给出下界证明及阈值常数。
We present two main theorems for stochastic processes with a Markov regime-switching model. First, we work on an existence-uniqueness theorem for a Stochastic Differential Delay Equation with Jumps and Regimes (SDDEJRs). Then, we provide the duality between an SDDEJR and an Anticipated Backward Stochastic Differential Equation with Jumps and Regimes (ABSDEJRs). Our goal is to provide two technical and basis theorems for the future theoretical and applied developments of time-delayed and time-advanced models.
The central result of this paper is an analytic duality relation for real-valued Lévy processes killed upon exiting a half-line. By Nagasawa's theorem, this yields a remarkable time-reversal identity involving the Lévy process conditioned to stay positive. As examples of applications, we construct a version of the Lévy process indexed by the entire real line and started from $-\infty$ which enjoys a natural spatial-stationarity property, and point out that the latter leads to a natural Lamperti-type representation for self-similar Markov processes in $(0,\infty)$ started from the entrance poin…
We introduce a new method, which we call stochastic fusion, which takes an exclusion process and constructs an interacting particle systems in which more than one particle may occupy a lattice site. The construction only requires the existence of stationary measures of the original exclusion process on a finite lattice. If the original exclusion process satisfies Markov duality on a finite lattice, then the construction produces Markov duality functions (for some initial conditions) for the fused exclusion process. The stochastic fusion construction is based off of the Rogers-Pitman intertwini…
Representations of Hecke algebras and Markov dualities for interacting particle systems
本文对生成元由双点位等幂算子之和构成的连续时间 Markov 过程进行分类，揭示了其与 Hecke 代数不可约表示的联系，并通过标准的 Baxterisation 过程导出了不守恒粒子数系统的 Young-Baxter 方程新解。
Continuous-state branching processes with Lévy-Khintchine drift-interaction: Laplace duality and Fellerian extensions
本文研究了一类由Lévy-Khintchine型漂移驱动的连续状态分支过程（CBDI），通过拉普拉斯对偶性和Feller扩展分析了其边界行为和分支与漂移-相互作用机制的交换特性。
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