M. Neklyudov
In this article I will prove new representation for the Levi-Civita connection in terms of the stochastic flow corresponding to Brownian motion on manifold.
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Analogous to the characterisation of Brownian motion on a Riemannian manifold as the development of Brownian motion on a Euclidean space, we construct sub-Riemannian diffusions on equinilpotentisable sub-Riemannian manifolds by developing a canonical stochastic process arising as the lift of Brownian motion to an associated model space. The notion of stochastic development we introduce for equinilpotentisable sub-Riemannian manifolds uses Cartan connections, which take the place of the Levi-Civita connection in Riemannian geometry. We first derive a general expression for the generator of the …
Connections between a system of Forward-Backward SDEs and Backward Stochastic PDEs related to the utility maximiza- tion problem is established. Besides, we derive another version of FBSDE of the same problem and prove an existence of a solution
Let $G$ be a Lie Group with a left invariant connection such that its connection function is skew-symmetric. Our main goal is to show a version of Pluzhnikov's Theorem for this kind of connection. To this end, we use the stochastic logarithm. More exactly, the stochastic logarithm gives characterizations for Brownian motions and Martingales in $G$, and these characterzations are used to prove Pluzhnikov's Theorem.
In this paper, we study one-dimensional backward stochastic differential equation with jump under logarithmic growth assumption in the z-variable (|z|\sqrt{|\ln|z|}|) and an L^p terminal value (for a suitable p>2). We show the existence and the uniqueness of the solution when the noise is driven by a Brownian motion and an independent Poisson random measure. In addition, we highlight the connection of such BSDEs with stochastic optimal control problem, where we show the existence of an optimal strategy for the stochastic control problem.
Connections between coupling and Ishii-Lions methods for tug-of-war with noise stochastic games
本文梳理了带噪声拔河随机博弈中耦合方法与Ishii-Lions方法两种正则性方法的联系，分别应用于博弈与$p$-拉普拉斯方程粘性解。
A quantum Markov semigroup can be represented via classical diffusion processes solving a stochastic Schrödinger equation. In this paper we first prove that a quantum Markov semigroup is irreducible if and only if classical diffusion processes are total in the Hilbert space of the system. Then we study the relationship between irreducibility of a quantum Markov semigroup and properties of these diffusions such as accessibility, the Lie algebra rank condition, and irreducibility. We prove that all these properties are, in general, weaker than irreducibility of the quantum Markov semigroup, neve…
Following Le Jan and Watanabe we define a connection associated with a non-degenrrate diffusion operators. This connection is characterized here and shown to be the Levi-Civita connection for gradient systems. This both explains why such systems have useful properties and allows us to extend these properties to more general systems. Topics described here include: moment estimates for $Tξ_t$, a Weitzenböck formula for the generator of the semigroup on p-forms induced by the flow, a Bismut type formula for $d\log p_t$ in terms of an arbitrary metric connection, and a generalized Bochner vanishin…
We prove that a Gaussian ensemble of smooth random sections of a real vector bundle over compact manifold canonically defines a metric on the bundle together with a connection compatible with it. Additionally, we prove a refined Gauss-Bonnet-Chern theorem stating that if the bundle and the manifold are oriented, then the Euler form of the above connection can be identified, as a current, with the expectation of the random current defined by the zero-locus of a random section in the above Gaussian ensemble.
Tail Asymptotics of the Signature of various stochastic processes and its connection to the Quadratic Variation
本文扩展了布朗运动签名与二次变差联系的研究，证明了半鞅Itô签名与流逝时间及分数布朗运动fWIS签名与Hurst参数的联系，并提出了多维扩展猜想。
We give a new representation of fractional Brownian motion with Hurst parameter H<=1/2 using stochastic partial differential equations. This representation allows us to use the Markov property and time reversal, tools which are not usually available for fractional Brownian motion. We then give simple proofs that fractional Brownian motion does not hit points in the critical dimension, and that it does not have double points in the critical dimension. These facts were already known, but our proofs are quite simple and use some ideas of Levy.
Dynamical relationship between CAR algebras and determinantal point processes: point processes at finite temperature and stochastically positive KMS systems
本文拓展行列式点过程与CAR代数的动力学关系研究，提出KMS态相关随机过程理论的新应用，建立点配置空间平稳过程的统一代数理论，还推导了时空关联行列式公式并分析过程极限行为。
This two-part paper addresses the design of retail electricity tariffs for distribution systems with distributed energy resources such as solar power and storage. In particular, the optimal design of dynamic two-part tariffs for a regulated monopolistic retailer is considered, where the retailer faces exogenous wholesale electricity prices and fixed costs on the one hand and stochastic demands with inter-temporal price dependencies on the other. Part I presents a general framework and analysis for revenue adequate retail tariffs with advanced notification, dynamic prices and uniform connection…
A note on stochastic integrals as $L^2$-curves
本文完善了随机积分与Itô积分的联系，并探讨了其在随机偏微分方程中的应用。
The mean-field limit of a Markovian model describing the interaction of several classes of permanent connections in a network is analyzed. Each of the connections has a self-adaptive behavior in that its transmission rate along its route depends on the level of congestion of the nodes of the route. Since several classes of connections going through the nodes of the network are considered, an original mean-field result in a multi-class context is established. It is shown that, as the number of connections goes to infinity, the behavior of the different classes of connections can be represented …
Both stochastic and PDE modeling approaches have been used and compared in various context in biology. Typically, stochastic models are easier to parameterize, can be used to integrate underlying biological phenomena, but hard to analyze mathematically and usually computational expensive. PDE models are amenable to mathematical analysis and computational methods are well-developed, but biological processes are usually lumped together in them and parameters harder to obtain. Therefore it is important to find connections of different approaches. In this note, I give some investigation of methods…
In this article we propose a model for stochastic delay differential equation with jumps (SDDEJ) in a differentiable manifold $M$ endowed with a connection $\nabla$. In our model, the continuous part is driven by vector fields with a fixed delay and the jumps are assumed to come from a distinct source of (càdlàg) noise, without delay. The jumps occur along adopted differentiable curves with some dynamical relevance (with fictitious time) which allow to take parallel transport along them. Using a geometrical approach, in the last section, we show that the horizontal lift of the solution of an S…
In this paper, by virtue of Malliavin calculus, we establish a relationship between backward doubly stochastic differential equations with random coefficients and quasilinear stochastic PDEs, and thus extend the well-known nonlinear stochastic Feynman-Kac formula of Pardoux and Peng [14] to non-Markovian case.
Interacting Stochastic Systems with Agent-Based Sampling and Connection-Based Reinforcement
该研究提出一类基于采样与强化网络的交互随机系统，建模为随机近似方案，推导收敛准则、极限性质并证明其涵盖多种经典模型。
The goal of this paper is to develop a theory of graphon-valued stochastic processes, and to construct and analyse a natural class of such processes arising from population genetics. We consider finite populations where individuals change type according to Wright-Fisher resampling. At any time, each pair of individuals is linked by an edge with a probability that is given by a type-connection matrix, whose entries depend on the current empirical type distribution of the entire population via a fitness function. We show that, in the large-population-size limit and with an appropriate scaling of…
This paper is concerned with a characterization of the observability for a continuous-time hidden Markov model where the state evolves as a general continuous-time Markov process and the observation process is modeled as nonlinear function of the state corrupted by the Gaussian measurement noise. The main technical tool is based on the recently discovered duality relationship between minimum variance estimation and stochastic optimal control: The observability is defined as a dual of the controllability for a certain backward stochastic differential equation. Based on the dual formulation, a t…
In this paper, we consider the composition of two independent processes : one process corresponds to position and the other one to time. Such processes will be called iterated processes. We first propose an algorithm based on the Euler scheme to simulate the trajectories of the corresponding iterated processes on a fixed time interval. This algorithm is natural and can be implemented easily. We show that it converges almost surely, uniformly in time, with a rate of convergence of order 1/4 and propose an estimation of the error. We then extend the well known Feynman- Kac formula which gives a …
Typically, a stochastic model relates stochastic "inputs" and, perhaps, controls to stochastic "outputs". A general version of the Yamada-Watanabe and Engelbert theorems relating existence and uniqueness of weak and strong solutions of stochastic equations is given in this context. A notion of {\em compatibility} between inputs and outputs is critical in relating the general result to its classical forebears. The relationship between the compatibility condition and the usual formulation of stochastic differential equations driven by semimartingales is discussed.
This work is concerned with model reduction of stochastic differential equations and builds on the idea of replacing drift and noise coefficients of preselected relevant, e.g. slow variables by their conditional expectations. We extend recent results by Legoll & Lelièvre [Nonlinearity 23, 2131, 2010] and Duong et al. [Nonlinearity 31, 4517, 2018] on effective reversible dynamics by conditional expectations to the setting of general non-reversible processes with non-constant diffusion coefficient. We prove relative entropy and Wasserstein error estimates for the difference between the time marg…
We prove an identity on Hermitian random matrix models with external source relating the high rank cases to the rank 1 cases. This identity was proved and used in a previous paper of ours to study the asymptotics of the top eigenvalues. In this paper, we give an alternative, more conceptual proof of this identity based on a connection between the Hermitian matrix models with external source and the discrete KP hierarchy. This connection is obtained using the vertex operator method of Adler and van Moerbeke. The desired identity then follows from the Fay-like identity of the discrete KP tau vec…
We introduce the class of vector measures of bounded $γ$-variation and study its relationship with vector-valued stochastic integrals with respect to Brownian motions.
Tight estimates of exit/containment probabilities are of particular importance in many control problems. Yet, estimating the exit/containment probabilities is non-trivial: even for linear systems (Ornstein-Uhlenbeck processes), the containment probability can be computed exactly for only some particular values of the system parameters. In this paper, we derive tight bounds on the containment probability for a class of nonlinear stochastic systems. The core idea is to compare the "pull strength" (how hard the deterministic part of the system dynamics pulls towards the origin) experienced by the…
In this paper we study the relationship between functional forward-backward stochastic systems and path-dependent PDEs. In the framework of functional Itô calculus, we introduce a path-dependent PDE and prove that its solution is uniquely determined by a functional forward-backward stochastic system.
We prove a positivity theorem for a certain family of operators defined in terms of the stochastic six-vertex model. We explore connections of this result with other vertex models and $t$-deformed Schensted insertions.
Concentration bounds for stochastic systems with singular kernels
本文研究了具有奇异成对相互作用的弱相互作用随机粒子系统，通过建立变分上界，证明了混沌的受控熵传播意味着经验测度的指数集中界，从而推断出一类奇异随机系统的集中界。
In this paper, we discuss the relationships between stability and almost periodicity for solutions of stochastic differential equations. Our essential idea is to get stability of solutions or systems by some inherited properties of Lyapunov functions. Under suitable conditions besides Lyapunov functions, we obtain the existence of almost periodic solutions in distribution.
In this paper, a connection between bi-free probability and the theory of non-commutative stochastic processes is examined. Specifically it is demonstrated that the transition operators for non-commutative stochastic processes can be modelled using technology from bi-free probability. Several important examples are recovered with this approach and new formula are obtained for processes with free increments. The benefits of this approach are also discussed.
We review some recent results on connections between Brownian motion, Whittaker functions, random matrices and representation theory.
Percolation in the Stochastic Block Model
本文研究社区结构随网络规模演化的随机块模型序列中的渗流，发现临界平均度通常不等于1，并提出了社区尺度分支过程方法以准确预测相变，扩展了渗流研究至更现实场景。
We consider multi-class single-server queueing networks that have a product form stationary distribution. A new limit result proves a sequence of such networks converges weakly to a stochastic flow level model. The stochastic flow level model found is insensitive. A large deviation principle for the stationary distribution of these multi-class queueing networks is also found. Its rate function has a dual form that coincides with proportional fairness. We then give the first rigorous proof that the stationary throughput of a multi-class single-server queueing network converges to a proportional…
We consider a very general stochastic model for an SIR epidemic on a network which allows an individual's infectious period, and the time it takes to contact each of its neighbours after becoming infected, to be correlated. We write down the message passing system of equations for this model and prove, for the first time, that it has a unique feasible solution. We also generalise an earlier result by proving that this solution provides a rigorous upper bound for the expected epidemic size (cumulative number of infection events) at any fixed time $t>0$. We specialise these results to a homogene…
The present work demonstrates the connections between the Burgers, diffusion, and Schroedinger's equations. The starting point is a formulation of the stochastic mechanics, which is modeled along the lines of the scale relativity theory. The resulting statistical description obeys the Fokker-Planck equation. This paper further demonstrates the connection between the two approaches, embodied by the study of the Burgers equation, which from this perspective appears as a stochastic geodesic equation. The main result of the article is the transparent derivation of the Born rule from the starting p…
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