Eugene Lytvynov
We review the recent results on the Jacobi field of a (real-valued) Lévy process defined on a Riemannian manifold. In the case where the Lévy process is neither Gaussian, nor Poisson, the corresponding Jacobi field acts in an extended Fock space. We also give a unitary equivalent representation of the Jacobi field in a usual Fock space. This representation is inspired by a result by Accardi, Franz, and Skeide (2002).
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We derive an explicit formula for the Jacobi field that is acting in an extended Fock space and corresponds to an ($\R$-valued) Lévy process on a Riemannian manifold. The support of the measure of jumps in the Lévy--Khintchine representation for the Lévy process is supposed to have an infinite number of points. We characterize the gamma, Pascal, and Meixner processes as the only Lévy processes whose Jacobi field leaves the set of finite continuous elements of the extended Fock space invariant.
We define and study a multidimensional process that generalizes the eigenvalues of matrix Jacobi processes on the one hand and whose stationary distribution is given by the beta Jacobi ensemble on the other hand.
We generalize a previous result concerning free martingale polynomials for the stationary free Jacobi process of parameters $λ\in ]0.1], θ= 1/2$. Hopelessly, apart from the case $λ= 1$, the polynomials we derive are no longer orthogonal with respect to the spectral measure. As a matter of fact, we use the multiplicative renormalization to write down the corresponding orthogonality measure.
We establish uniqueness for a class of first-order Hamilton-Jacobi equations with Hamiltonians that arise from the large deviations of the empirical measure and empirical flux pair of weakly interacting Markov jump processes. As a corollary we obtain a large deviation principle for the trajectory of the empirical measure and empirical flux pair of such processes. As a second corollary we get same result in the setting where the jump-rates are time-periodic, with period-length that decreases to 0.
We study the Hamilton-Jacobi equation f - lambda Hf = h, where H f = e^{-f}Ae^f and where A is an operator that corresponds to a well-posed martingale problem. We identify an operator that gives viscosity solutions to the Hamilton-Jacobi equation, and which can therefore be interpreted as the resolvent of H. The operator is given in terms of optimization problem where the running cost is a path-space relative entropy. Finally, we use the resolvents to give a new proof of the abstract large deviation result of Feng and Kurtz.
We study $β$-Jacobi diffusion processes on alcoves in $\mathbb R^N$, depending on 3 parameters. Using elementary symmetric functions, we present space-time-harmonic functions and martingales for these processes $(X_t)_{t\ge0}$ which are independent from one parameter. This leads to a formula for $\mathbb E(\prod_{i=1}^N (y-X_{t,i}))$ in terms of classical Jacobi polynomials. For $t\to\infty$ this yields a corresponding formula for Jacobi ensembles and thus Aomoto's Selberg integral.
We study a family of quasi-birth-and-death (QBD) processes associated with the so-called first family of Jacobi-Koornwinder bivariate polynomials. These polynomials are orthogonal on a bounded region typically known as the swallow tail. We will explicitly compute the coefficients of the three-term recurrence relations generated by these QBD polynomials and study the conditions under we can produce families of discrete-time QBD processes. Finally, we show an urn model associated with one special case of these QBD processes.
We study the scaling limit of the spectrum of the β-Jacobi ensemble at the soft-edge and hard-edge for general values of β. We show that the limiting point processes correspond respectively to the stochastic Airy and Bessel point processes introduced in Ramirez-Rider-Virag and Ramirez-Rider.
We investigate the long time behavior of weakly dissipative semilinear Hamilton-Jacobi-Bellman (HJB) equations and the turnpike property for the corresponding stochastic control problems. To this aim, we develop a probabilistic approach based on a variant of coupling by reflection adapted to the study of controlled diffusion processes. We prove existence and uniqueness of solutions for the ergodic Hamilton-Jacobi-Bellman equation and different kind of quantitative exponential convergence results at the level of the value function, of the optimal controls and of the optimal processes. Moreover,…
We prove pathwise large-deviation principles of switching Markov processes by exploiting the connection to associated Hamilton-Jacobi equations, following Jin Feng's and Thomas Kurtz's method. In the limit that we consider, we show how the large-deviation problem in path-space reduces to a spectral problem of finding principal eigenvalues. The large-deviation rate functions are given in action-integral form. As an application, we demonstrate how macroscopic transport properties of stochastic models of molecular motors can be deduced from an associated principal-eigenvalue problem. The precise …
On The Cutoff Phenomenon For Dyson-Jacobi Processes
本文研究了Dyson-Jacobi过程的截止现象，通过结合黎曼几何与变形表示，给出了混合时间的显式公式，并提供了精确的定量估计。
The purpose of this paper is to study the existence and uniqueness of solutions to a system of Stochastic Differential Equations (SDEs). The coordinates are bounded by zero and one, and repulse each other according to a Coulombian like interaction force. We show the existence of strong and pathwise unique solutions to the system until the first multiple collision at zero or one, and give a sufficient condition on the parameters of the SDEs for this multiple collision not to occur in finite time.
Jacobi processes with jumps as neuronal models: a first passage time analysis
该研究提出带向下跳跃的马尔可夫雅可比过程作为神经元模型，通过分析其首次通过时间研究峰峰间期，推导拉普拉斯变换与期望闭式解，并开展神经元放电率数值研究。
Non-local Hamilton-Jacobi-Bellman equations for the stochastic optimal control of path-dependent piecewise deterministic processes
本文研究路径依赖分段确定性过程的最优控制，建立了动态规划原理，并证明值函数为非局部路径依赖HJB方程的唯一极小极大解，首次给出此类方程非光滑解的适定性结果。
We study the asymptotics of the global fluctuations for the difference between two adjacent levels in the $β$--Jacobi corners process (multilevel and general $β$ extension of the classical Jacobi ensemble of random matrices). The limit is identified with the derivative of the $2d$ Gaussian Free Field. Our main tools are integral forms for the (Macdonald-type) difference operators originating from the shuffle algebra.
Finite free probability and $S$ transforms of Jacobi processes
本文研究了雅可比过程在自由和有限自由概率理论中的S变换，推导了相关偏微分方程，并证明了冻结雅可比过程在高维下的收敛性。
We present a unified approach to a couple of central limit theorems for radial random walks on hyperbolic spaces and time-homogeneous Markov chains on the positive half line whose transition probabilities are defined in terms of the Jacobi convolutions. The proofs of all results are based on limit results for the associated Jacobi functions. In particular, we consider the cases where the first parameter (i.e., the dimension of the hyperbolic space) tends to infinity as well as the cases $ϕ_{iρ-λ}^{(α,β)}(t)$ for small $λ$, and $ϕ_{iρ-nλ}^{(α,β)}(t/n)$ for $n\to\infty$. The proofs of all these …
We show that for $β\ge 1$ the semigroups of $β$-Laguerre and $β$-Jacobi processes of different dimensions are intertwined in analogy to a similar result for $β$-Dyson Brownian motion recently obtained by Ramanan and Shkolnikov. These intertwining relations generalize to arbitrary $β\ge 1$ the ones obtained for $β=2$ by the author, O'Connell and Warren between $h$-transformed Karlin-McGregor semigroups. Moreover they form the key step towards constructing a multilevel process in a Gelfand-Tsetlin pattern leaving certain Gibbs measures invariant. Finally as a by product we obtain a relation betw…
We prove the large deviation principle for the trajectory of a broad class of mean field interacting Markov jump processes via a general analytic approach based on viscosity solutions. Examples include generalized Ehrenfest models as well as Curie-Weiss spin flip dynamics with singular jump rates. The main step in the proof of the large deviation principle, which is of independent interest, is the proof of the comparison principle for an associated collection of Hamilton-Jacobi equations. Additionally, we show that the large deviation principle provides a general method to identify a Lyapunov …
We study Markov chains formed by squared singular values of products of truncated orthogonal, unitary, symplectic matrices (corresponding to the Dyson index $β= 1,2,4$ respectively) where time corresponds to the number of terms in the product. More generally, we consider the $β$-Jacobi product process obtained by extrapolating to arbitrary $β> 0$. When the time scaling is preserved, we show that the global fluctuations are jointly Gaussian with explicit covariances. For time growing linearly with matrix size, we show convergence of moments after suitable rescaling. When $β= 2$, our results imp…
The spectral measures of random Jacobi matrices related to beta ensembles at high temperature and Dirichlet processes
本文研究了高温下β集合随机雅可比矩阵的谱测度，证明其收敛于狄利克雷过程并提供显式谱测度的例子。
Infinite-dimensional $q$-Jacobi Markov processes
本文构造了 $N$ 维 $q$-Jacobi 马尔可夫过程，并证明其在 $N\to\infty$ 时存在极限，得到无限维 Feller 过程，其证明依赖于大 $q$-Jacobi 多项式的对称函数提升。
We introduce a stochastic dynamics related to the measures that arise in harmonic analysis on the infinite-dimensional unitary group. Our dynamics is obtained as a limit of a sequence of natural Markov chains on Gelfand-Tsetlin graph. We compute finite-dimensional distributions of the limit Markov process, the generator and eigenfunctions of the semigroup related to this process. The limit process can be identified with Doob h-transform of a family of independent diffusions. Space-time correlation functions of the limit process have a determinantal form.
The stochastic Jacobi flow
本文研究实数轴上布朗环汤的占据场条件化问题，构造了雅可比流，并统一了相关分解理论，证明了与Bass-Burdzy流的联系。
We investigate a simple velocity jump process in the regime of large deviation asymptotics. New velocities are taken randomly at a constant, large, rate from a Gaussian distribution with vanishing variance. The Kolmogorov forward equation associated with this process is the linear BGK kinetic transport equation. We derive a new type of Hamilton-Jacobi equation which is nonlocal with respect to the velocity variable. We introduce a suitable notion of viscosity solution, and we prove well-posedness in the viscosity sense. We also prove convergence of the logarithmic transformation towards this l…
Open Markets and Hybrid Jacobi Processes
该研究提出统一方法，结合开放市场与混合雅可比过程模型族，分析随机投资组合理论相关问题，推导市场特性并放宽开放市场定义以提升分析可处理性。
Large deviation for Markov processes can be studied by Hamilton--Jacobi equation techniques. The method of proof involves three steps: First, we apply a nonlinear transform to generators of the Markov processes, and verify that limit of the transformed generators exists. Such limit induces a Hamilton--Jacobi equation. Second, we show that a strong form of uniqueness (the comparison principle) holds for the limit equation. Finally, we verify an exponential compact containment estimate. The large deviation principle then follows from the above three verifications. This paper illustrates such a m…
We consider the family of nearest neighbour interacting particle systems on $\mathbb{Z}$ allowing $0$, $1$ or $2$ particles at a site. We parametrize a wide subfamily of processes exhibiting product blocking measure and show how this family can be "stood up" in the sense of Balázs and Bowen (2018). By comparing measures we prove new three variable Jacobi style identities, related to counting certain generalised Frobenius partitions with a $2$-repetition condition. By specialising to specific processes we produce two variable identities that are shown to relate to Jacobi triple product and vari…
In [Yu.M. Berezansky, E. Lytvynov, D. A. Mierzejewski, Ukrainian Math. J. 55 (2003), 853--858 ], the Jacobi field of a Lévy process was derived. This field consists of commuting self-adjoint operators acting in an extended (interacting) Fock space. However, these operators have a quite complicated structure. In this note, using ideas from [L. Accardi. U. Franz, M. Skeide, Comm. Math. Phys. 228 (2002), 123--150] and [E. Lytvynov, Infin. Dimen. Anal. Quant. Prob. Rel. Top. 7 (2004), 619--629], we obtain a unitary equivalent representation of the Jacobi field of a Lévy process. In this representa…
Large deviations for Markov processes with switching and homogenisation via Hamilton-Jacobi-Bellman equations
本文针对带内状态切换的分子马达扩散模型，通过分析同时依赖位置与动量的Hamilton–Jacobi–Bellman方程，用Feng-Kurtz方法证明了齐次化极限下轨迹的大偏差原理。
Wigner- and Marchenko-Pastur-type limits for Jacobi processes
本文研究 BC 型根系驱动的 Jacobi 过程，利用随机微分方程推导出当粒子数趋于无穷时经验分布的 Wigner 半圆律和 Marchenko-Pastur 律的几乎必然极限，并推广了 Jacobi 系综及多项式零点的相关平稳极限结果。
We define Laguerre and Jacobi analogues of the Warren process. That is, we construct local dynamics on a triangular array of particles so that the projections to each level recover the Laguerre and Jacobi eigenvalue processes of König-O'Connell and Doumerc and the fixed time distributions recover the joint distribution of eigenvalues in multilevel Laguerre and Jacobi random matrix ensembles. Our techniques extend and generalize the framework of intertwining diffusions developed by Pal-Shkolnikov. One consequence is the construction of particle systems with local interactions whose fixed time d…
We review several applications of Berezansky's projection spectral theorem to Jacobi fields in a symmetric Fock space, which lead to Lévy white noise measures.
For arbitrary $β> 0$, we use the orthogonal polynomials techniques developed by R. Killip and I. Nenciu to study certain linear statistics associated with the circular and Jacobi $β$ ensembles. We identify the distribution of these statistics then prove a joint central limit theorem. In the circular case, similar statements have been proved using different methods by a number of authors. In the Jacobi case these results are new.
We identify the representation of the square of white noise obtained by L. Accardi, U. Franz and M. Skeide in [Comm. Math. Phys. 228 (2002), 123--150] with the Jacobi field of a Lévy process of Meixner's type.
Modeling Transmission Intensity in SI Epidemics via CIR and Jacobi Processes: Asymptotic Results and Preliminary Intervention Strategies
本文通过CIR和雅可比过程建模SI传染病的传播强度，研究其渐近行为及非干预和成功干预策略下的传播动态。
A Viscosity Solution Theory of Stochastic Hamilton-Jacobi-Bellman equations in the Wasserstein Space
本文研究了Wasserstein空间中允许随机系数的非马尔可夫平均场型控制问题的随机Hamilton-Jacobi-Bellman方程，证明了值函数是其唯一粘性解，并克服了空间局部紧性缺失与非马尔可夫设定的挑战。
Using probabilistic methods we study the existence of viscosity solutions to non-linear integro-differential equations $$\partial_t u(t,x) - \sup_{α\in I} \bigg( b_α(x) \cdot \nabla_x u(t,x) + \frac{1}{2} \text{tr}\left(Q_α(x) \cdot \nabla^2_x u(t,x)\right) +\int_{y \neq 0} \big(u(t,x+y)-u(t,x)-\nabla_x u(t,x) \cdot h(y) \big) \, ν_α(x,dy) \bigg) = 0$$ with initial condition $u(0,x)= φ(x)$; here $(b_α(x),Q_α(x),ν_α(x,dy))$, $α\in I$, $x \in \mathbb{R}^d$, is a family of Lévy triplets and $h$ is some truncation function. The solutions, which we construct, are of the form $u(t,x) = T_t φ(x)$ for…
We prove that the two-dimensional Gaussian Free Field describes the asymptotics of global fluctuations of a multilevel extension of the general beta Jacobi random matrix ensembles. Our approach is based on the connection of the Jacobi ensembles to a degeneration of the Macdonald processes that parallels the degeneration of the Macdonald polynomials to to the Heckman-Opdam hypergeometric functions (of type A). We also discuss the beta goes to infinity limit.
We argue that Hamilton-Jacobi equations provide a convenient and intuitive approach for studying the large-scale behavior of mean-field disordered systems. This point of view is illustrated on the problem of inference of a rank-one matrix. We compute the large-scale limit of the free energy by showing that it satisfies an approximate Hamilton-Jacobi equation with asymptotically vanishing viscosity parameter and error term.
A mean field Jacobi process for modeling sustainable tourism
本文提出均场雅可比过程用于建模可持续旅游业，通过随机控制框架在不确定性下设计可持续旅游业，并探讨其在不同参数下的应用与结果。
The paper is concerned with the deterministic limit of mean field games with the nonlocal coupling. It is assumed that the dynamics of mean field games are given by nonlinear Markov processes. This type of games includes stochastic mean field games as well as mean field games with finite state space. We consider the limiting deterministic mean field game within the framework of minimax approach. The concept of minimax solutions is close to the probabilistic formulation. In this case the Hamilton--Jacobi equation is considered in the minimax/viscosity sense, whereas the flow of probabilities is…
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