Z. -Q. Chen, P. J. Fitzsimmons, K. Kuwae, T. -S. Zhang
Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an Itô formula for Dirichlet processes is obtained.
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Nakao's stochastic integrals for continuous additive functionals of zero energy are extended from the symmetric Dirichlet forms setting to the non-symmetric Dirichlet forms setting. Ito's formula in terms of the extended stochastic integrals is obtained.
This erratum corrects the article arXiv:0806.2044 published in Ann. Probab. 36 (2008) 931--970
We refine stochastic calculus for symmetric Markov processes without using time reverse operators. Under some conditions on the jump functions of locally square integrable martingale additive functionals, we extend Nakao's divergence-like continuous additive functional of zero energy and the stochastic integral with respect to it under the law for quasi-everywhere starting points, which are refinements of the previous results under the law for almost everywhere starting points. This refinement of stochastic calculus enables us to establish a generalized Fukushima decomposition for a certain cl…
We describe stochastic calculus in the context of processes that are driven by an adapted point process of locally finite intensity and are differentiable between jumps. This includes Markov chains as well as non-Markov processes. By analogy with Itô processes we define the drift and diffusivity, which we then use to describe a general sample path estimate. We then give several examples, including ODE approximation, processes with linear drift, first passage times, and an application to the stochastic logistic model.
Let $(\mathcal{E},D(\mathcal{E}))$ be a quasi-regular semi-Dirichlet form and $(X_t)_{t\geq0}$ be the associated Markov process. For $u\in D(\mathcal{E})_{loc}$, denote $A_t^{[u]}:=\tilde{u}(X_{t})-\tilde{u}(X_{0})$ and $F^{[u]}_t:=\sum_{0<s\leq t}(\tilde u(X_{s})-\tilde u(X_{s-}))1_{\{|\tilde u(X_{s})-\tilde u(X_{s-})|>1\}}$, where $\tilde{u}$ is a quasi-continuous version of $u$. We show that there exist a unique locally square integrable martingale additive functional $Y^{[u]}$ and a unique continuous local additive functional $Z^{[u]}$ of zero quadratic variation such that $$A_t^{[u]}=Y_t^…
We introduce a variational theory for processes adapted to the multi-dimensional Brownian motion filtration. The theory provides a differential structure which describes the infinitesimal evolution of Wiener functionals at very small scales. The Markov property is replaced by a finite-dimensional approximation procedure based on controlled inter-arrival times and jumps of approximating martingales. The theory reveals that a large class of adapted processes follow a differential rule which is similar in nature to a fundamental theorem of calculus in the context of Wiener functionals. Null stoch…
Elements of the Stochastic Calculus for a Class of Boltzmann type processes and Application to Regularity of Densities
该研究针对一类分段确定性马尔可夫过程，提出经典马利亚万微积分的非高斯对应随机微积分方法，并将其应用于探究N粒子玻尔兹曼型过程的密度正则性。
A method for computing integrals of polynomial functions on compact symmetric spaces is given. Those integrals are expressed as sums of functions on symmetric groups.
A calculus for Markov chain Monte Carlo: studying approximations in algorithms
本文提出马尔可夫链蒙特卡洛微积分，通过定义 MCMC 核关于不变分布的导数及均值不等式，为分析和证明使用不变分布近似的交互式与序贯 MCMC 算法收敛性提供统一框架。
Malliavin calculus for marked binomial processes: portfolio optimisation in the trinomial model and compound Poisson approximation
本文构建带标记二项过程的Markov-Malliavin结构与混沌展开体系，将其应用于复合泊松近似的Chen-Stein方法重构及三叉树模型投资组合优化研究。
Chen, Fitzsimmons, Kuwae and Zhang (Ann. Probab. 36 (2008) 931-970) have established an Ito formula consisting in the development of F(u(X)) for a symmetric Markov process X, a function u in the Dirichlet space of X and any $\mathcal{C}^2$-function F. We give here an extension of this formula for u locally in the Dirichlet space of X and F admitting a locally bounded Radon-Nikodym derivative. This formula has some analogies with various extended Ito formulas for semi-martingales using the local time stochastic calculus. But here the part of the local time is played by a process $(Γ^a_t,a\in \m…
The purpose of this paper is to consider the exit-time problem for a finite-range Markov jump process, i.e, the distance the particle can jump is bounded independent of its location. Such jump diffusions are expedient models for anomalous transport exhibiting super-diffusion or nonstandard normal diffusion. We refer to the associated deterministic equation as a volume-constrained nonlocal diffusion equation. The volume constraint is the nonlocal analogue of a boundary condition necessary to demonstrate that the nonlocal diffusion equation is well-posed and is consistent with the jump process. …
In this paper, we investigate spectral properties of explosive symmetric Markov processes. Under a condition on its life time, we prove the $L^1$-semigroup of Markov processes become compact operators.
We give some relationships between the first Dirichlet eigenvalues and the exit time moments for the general symmetric Markov processes. As applications, we present some examples, including symmetric diffusions and $α$-stable processes, and provide the estimates of their first Dirichlet eigenvalues and the exit time moments.
Let $X$ be an irreducible symmetric Markov process with the strong Feller property. We assume, in addition, that $X$ is explosive and has a tightness property. We then prove the existence and uniqueness of quasi-stationary distributions of $X$.
This article discusses the usage of a partiton based Fubini calculus for Poisson processes. The approach is an amplification of Bayesian techniques developed in Lo and Weng for gamma/Dirichlet processes. Applications to models are considered which all fall within an inhomogeneous spatial extension of the size biased framework used in Perman, Pitman and Yor. Among some of the results; an explicit partition based calculus is then developed for such models, which also includes a series of important exponential change of measure formula. These results are applied to obtain results for Levy-Cox mod…
For each $n$ let $Y^n_t$ be a continuous time symmetric Markov chain with state space $n^{-1} \Z^d$. A condition in terms of the conductances is given for the convergence of the $Y^n_t$ to a symmetric Markov process $Y_t$ on $\R^d$. We have weak convergence of $\{Y^n_t: t\leq t_0\}$ for every $t_0$ and every starting point. The limit process $Y$ has a continuous part and may also have jumps.
Skellam compound Poisson approximation to the sums of symmetric Markov dependent random variables
研究对称马尔可夫相依三点随机变量之和的近似问题，用Skellam随机变量近似该和，并在局部、总变差和Wasserstein度量下估计精度，利用测度卷积性质完成证明。
By making full use of heat kernel estimates, we establish the integral tests on the zero-one laws of upper and lower bounds for the sample path ranges of symmetric Markov processes. In particular, these results concerning on upper rate bounds are applicable for local and non-local Dirichlet forms, while lower rate bounds are investigated in both subcritical setting and critical setting.
Hausdorff dimensions of inverse images and collision time sets for symmetric Markov processes
本文利用热核估计及 Hawkes、Jain–Pruitt 的方法，确定了一大类对称马尔可夫过程逆像与碰撞时间集的豪斯多夫维数，并应用于扩散、稳定型及带跳过程。
We study Girsanov's theorem in the context of symmetric Markov processes, extending earlier work of Fukushima-Takeda and Fitzsimmons on Girsanov transformations of ``gradient type.'' We investigate the most general Girsanov transformation leading to another symmetric Markov process. This investigation requires an extension of the forward-backward martingale method of Lyons-Zheng, to cover the case of processes with jumps.
Using the method of transportation-information inequality introduced in \cite{GLWY}, we establish Bernstein type's concentration inequalities for empirical means $\frac 1t \int_0^t g(X_s)ds$ where $g$ is a unbounded observable of the symmetric Markov process $(X_t)$. Three approaches are proposed : functional inequalities approach ; Lyapunov function method ; and an approach through the Lipschitzian norm of the solution to the Poisson equation. Several applications and examples are studied.
The spatial symmetry property of truncated birth-death processes studied in Di Crescenzo [6] is extended to a wider family of continuous-time Markov chains. We show that it yields simple expressions for first-passage-time densities and avoiding transition probabilities, and apply it to a bilateral birth-death process with jumps. It is finally proved that this symmetry property is preserved within the family of strongly similar Markov chains.
We consider symmetric Markov chains on $\Bbb Z^d$ where we do {\bf not} assume that the conductance between two points must be zero if the points are far apart. Under a uniform second moment condition on the conductances, we obtain upper bounds on the transition probabilities, estimates for exit time probabilities, and certain lower bounds on the transition probabilities. We show that a uniform Harnack inequality holds if an additional assumption is made, but that without this assumption such an inequality need not hold. We establish a central limit theorem giving conditions for a sequence of …
Stochastic Calculus for Pathwise Observables of Markov-Jump Processes: Unification of Diffusion and Jump Dynamics
本文通过建立马尔可夫跳跃过程路径观测的完整随机微积分，统一了扩散与跳跃动力学，推导了热力学不等式并讨论了饱和条件，同时将框架与量子 unraveling 和开放量子系统 Belavkin 方程联系起来。
This paper considers a controlled Itô-Lévy process where the information available to the controller is possibly less than the overall information. All the system coefficients and the objective performance functional are allowed to be random, possibly non-Markovian. Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjoint process is explicitly expressed.
Characterization of subordinate symmetric Markov processes
本文刻画了次对称马尔可夫过程的跳跃核，建立了其遵循Liu-Murugan的等价条件，并阐明了与扩散类型不同的尺度，结果可推广至非次过程。
We develop a stochastic analysis for a Gaussian process $X$ with singular covariance by an intrinsic procedure focusing on several examples such as covariance measure structure processes, bifractional Brownian motion, processes with stationary increments. We introduce some new spaces associated with the self-reproducing kernel space and we define the Paley-Wiener integral of first and second order even when $X$ is only a square integrable process continuous in $L^2$. If $X$ has stationary increments, we provide necessary and sufficient conditions so that its paths belong to the self-reproducin…
We consider a large class of symmetric pure jump Markov processes dominated by isotropic unimodal Lévy processes with weak scaling conditions. First, we establish sharp two-sided heat kernel estimates for these processes in $C^{1,1}$ open sets. As corollaries of our main results, we obtain sharp two-sided Green function estimates and a scale invariant boundary Harnack inequality with explicit decay rates in $C^{1,1}$ open sets.
The study of distributed order calculus usually concerns about fractional derivatives of the form $\int_0^1 \partial^αu \, m(dα)$ for some measure $m$, eventually a probability measure. In this paper an approach based on Lévy mixing is proposed. Non-decreasing Lévy processes associated to Lévy triplets of the form $ła(y), b(y), ν(ds, y) \r$ are considered and the parameter $y$ is randomized by means of a probability measure. The related subordinators are studied from different point of views. Some distributional properties are obtained and the interplay with inverse local times of Markov proce…
According to the theory of functional inequalities, a non-symmetric Markov semigroup has better properties than the corresponding symmetric one. For instance, there exist non-symmetric Markov semigroups which are hypercontractive (and thus converge exponentially in both $L^2$ and entropy), but the symmetric ones are even not ergodic. In this paper, we aim to search for reasonable conditions to ensure that a non-symmetric Markov semigroup and its symmetrization share the properties of exponential convergence, uniform integrability, hypercontractivity, and supercontractivity. Since in the symmet…
We study distributions of meeting times for finite symmetric Markov chains. For Markov kernels defined on large state spaces which satisfy certain weak inhomogeneity in return probabilities of points up to large numbers of steps, we obtain approximation, with explicit error bounds, of the Laplace transforms of some meeting times (without scaling) by ratios of Green functions closely related to hitting times of points. In studying this approximation, we identify some key matrix power series in Markov kernels weighted with solutions to a discrete transport-like equation with explicit coefficient…
Consider a symmetric unitary random matrix $V=(v_{ij})_{1 \le i,j \le N}$ from a circular orthogonal ensemble. In this paper, we study moments of a single entry $v_{ij}$. For a diagonal entry $v_{ii}$ we give the explicit values of the moments, and for an off-diagonal entry $v_{ij}$ we give leading and subleading terms in the asymptotic expansion with respect to a large matrix size $N$. Our technique is to apply the Weingarten calculus for a Haar-distributed unitary matrix.
In this paper, we establish sharp two-sided heat kernel estimates for a large class of symmetric Markov processes in exterior $C^{1,η}$ open sets for all $t> 0$. The processes are symmetric pure jump Markov processes with jumping kernel intensity $$κ(x, y)ψ(|x-y|)^{-1}|x-y|^{-d-α}$$ where $α\in(0,2)$, $ψ$ is an increasing function on $[ 0, \infty)$ with $ψ(r)=1$ on $0<r\le 1$ and $c_1e^{c_2r^β}\le ψ(r)\le c_3e^{c_4r^β}$ on $r>1$ for $β\in[0, \infty]$. A symmetric function $κ(x, y)$ is bounded by two positive constants and $|κ(x, y)-κ(x,x)|\le c_5 |x-y|^ρ$ for $|x-y|<1$ and $ρ>α/2$. As a coroll…
In this paper, we study sharp Dirichlet heat kernel estimates for a large class of symmetric Markov processes in $C^{1,η}$ open sets. The processes are symmetric pure jump Markov processes with jumping intensity $κ(x,y) ψ_1 (|x-y|)^{-1} |x-y|^{-d-α}$, where $α\in (0,2)$. Here, $ψ_1$ is an increasing function on $[ 0, \infty )$, with $ψ_1(r)=1$ on $0<r \le 1$ and $c_1e^{c_2r^β} \le ψ_1(r) \le c_3 e^{c_4r^β}$ on $r>1$ for $β\in [0,\infty]$, and $ κ( x, y)$ is a symmetric function confined between two positive constants, with $|κ(x,y)-κ(x,x)|\leq c_5|x-y|^ρ$ for $|x-y|<1$ and $ρ>α/2$. We establis…
We study a multivariate Markov chain on the symmetric group with remarkable enumerative properties. We conjecture that the stationary distribution of this Markov chain can be expressed in terms of positive sums of Schubert polynomials. This Markov chain is a multivariate generalization of a Markov chain introduced by the first author in the study of random affine Weyl group elements.
We show that the class of inductive limits of the representations of finite symmetric groups with simple spectrum coinsides with the class of Markov representations of the infinite symmetric group associated with Markov measures on the space of infinite Young tableaux. We also show that the representations of infinite symmetric group induced from identity representation of two-block Young subgroup are Markov representations and find explicit formulas for transition probabilities of corresponding Markov measure on the Young diagrmas. Induced two-row representations of finite symmetric group are…
Weingarten calculus for centered random permutation matrices
本文研究了中心随机排列矩阵在对称群S_N中的Weingarten算子，推导了中心情况下的公式及符号尊重公式，并揭示了Kummer的连贯超几何函数是该算子的基本构建块，为随机排列矩的代数和渐进行为提供了新的见解。
Markov chain approximations of symmetric jump processes are investigated. Tightness results and a central limit theorem are established. Moreover, given the generator of a symmetric jump process with state space $\mathbbm{R}^d$ the approximating Markov chains are constructed explicitly. As a byproduct we obtain a definition of the Sobolev space $H^{α/2}(\mathbbm{R}^d)$, $α\in (0,2)$, that is equivalent to the standard one.
In this paper, when a given symmetric Markov process X satisfies the stability of global heat kernel two-sided (upper) estimates by Markov perturbations, we give a necessary and sufficient condition on the stability of global two-sided (upper) estimates for fundamental solution of Feynman-Kac semigroup of X. As a corollary, under the same assumptions, a weak type global two-sided (upper) estimates holds for the fundamental solution of Feynman-Kac semigroup with (extended) Kato class conditions for measures. This generalizes all known results on the stability of global integral kernel estimates…
Solutions of stochastic Volterra (integral) equations are not Markov processes, and therefore classical methods, like dynamic programming, cannot be used to study optimal control problems for such equations. However, we show that by using {\em Malliavin calculus} it is possible to formulate a modified functional type of {\em maximum principle} suitable for such systems. This principle also applies to situations where the controller has only partial information available to base her decisions upon. We present both a sufficient and a necessary maximum principle of this type, and then we use the …
We study optimal investment strategies that maximize expected utility from consumption and terminal wealth in a pure-jump asset price model with Markov-modulated (regime switching) jump-size distributions. We give sufficient conditions for existence of optimal policies and find closed-form expressions for the optimal value function for agents with logarithmic and fractional power (CRRA) utility in the case of two-state Markov chains. The main tools are convex duality techniques, stochastic calculus for pure-jump processes and explicit formulae for the moments of telegraph processes with Markov…
It is well known that Nash-type inequalities for symmetric Dirichlet forms are equivalent to on-diagonal heat kernel upper bounds for the associated symmetric Markov semigroups. In this paper, we show that both imply (and hence are equivalent to) off-diagonal heat kernel upper bounds under some mild assumptions. Our approach is based on a new generalized Davies' method. Our results extend that of \cite{CKS} for Nash-type inequalities with power order considerably and also extend that of \cite{Gri} for second order differential operators on a complete non-compact manifold.
We establish the equivalence of the analytic and probabilistic notions of subharmonicity in the framework of general symmetric Hunt processes on locally compact separable metric spaces, extending an earlier work of the first named author on the equivalence of the analytic and probabilistic notions of harmonicity. As a corollary, we prove a strong maximum principle for locally bounded finely continuous subharmonic functions in the space of functions locally in the domain of the Dirichlet form under some natural conditions.
The approximation of integral functionals with respect to a stationary Markov process by a Riemann-sum estimator is studied. Stationarity and the functional calculus of the infinitesimal generator of the process are used to get a better understanding of the estimation error and to prove a general error bound. The presented approach admits general integrands and gives a unifying explanation for different rates obtained in the literature. Several examples demonstrate how the general bound can be related to well-known function spaces.
Given a (conservative) symmetric Markov process on a metric space we consider related bilinear forms that generalize the energy form for a particle in an electromagnetic field. We obtain one bilinear form by semigroup approximation and another, closed one, by using a Feynman-Kac-Itô formula. If the given process is Feller, its energy measures have densities and its jump measure has a kernel, then the two forms agree on a core and the second is a closed extension of the first. In this case we provide the explicit form of the associated Hamiltonian.
We study model embeddability, which is a variation of the famous embedding problem in probability theory, when apart from the requirement that the Markov matrix is the matrix exponential of a rate matrix, we additionally ask that the rate matrix follows the model structure. We provide a characterisation of model embeddable Markov matrices corresponding to symmetric group-based phylogenetic models. In particular, we provide necessary and sufficient conditions in terms of the eigenvalues of symmetric group-based matrices. To showcase our main result on model embeddability, we provide an applicat…
Quantum Reverse Hypercontractivity for KMS-Symmetric Quantum Markov Semigroups
本文研究KMS对称量子马尔可夫半群的反向超压缩性，构造了阻碍正一致率的例子，并证明在特定参数区间内所有本原半群满足该性质，且对图情形给出结果。
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