Hongjie Dong, N. V. Krylov
Time-inhomogeneous controlled diffusion processes in both cylindrical and noncylindrical domains are considered. Bellman's principle and its applications to proving the continuity of value functions are investigated.
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In this article, we consider time-inhomogeneous diffusive particle systems, whose particles jump from the boundary of a bounded open subset of $\R^d$, $d\geq 1$. We give a sufficient criterion for the family of empirical distributions of such systems to be uniformly tight, independently of the jump location of the particles. As an application, we show that the conditional distribution of a family of time-inhomogeneous and environment-dependent diffusions conditioned not to hit the boundary of a bounded open subset of $\R^d$ is uniformly tight.
We prove strong existense of solutions of Itô's stochastic time dependent equations with irregular diffusion and drift terms of Morrey class type.
We consider a time inhomogeneous Cox-Ingersoll-Ross diffusion with positive jumps. We exploit a branching property to prove existence of a unique strong solution under a restrictive condition on the jump measure. We give Laplace transforms for the transition probabilities, with an interpretation in terms of limits of mixtures over Gamma laws.
We prove strong existence and uniqueness of solutions of Itô's stochastic time dependent equations with irregular diffusion and drift terms of Morrey class type. In a sense we are treating a "supercritical" case.
In this paper we will make the computation of the Onsager-Machlup functional of an inhomogeneous uniformly elliptic diffusion process. This functional will have formally the same picture as in the homogeneous case, the only difference come from the infinitesimal variation of the volume. For example in the Ricci flow case, we find some functional which is not so far to the $\mathcal L_{0} $ distance used by Lott to study this flow \cite{Lott:08}. We finish by a application to small ball probability for weighted sup norm, for inhomogeneous diffusion.
Once again on weak solutions of time inhomogeneous Itô's equations with VMO diffusion and Morrey drift
该研究针对带非正则扩散项和混合范数Morrey类漂移项的伊藤时变随机方程，证明了其弱解的存在性与弱唯一性。
On strong solutions of time inhomogeneous Itô's equations with Morrey diffusion gradient and drift. A supercritical case
本文在超临界情形下研究时间非齐次Itô随机方程，证明了当扩散梯度和漂移项属于Morrey空间且不规则时强解的存在性，并讨论了强唯一性。
We study a kinetic stochastic model with a non-linear time-inhomogeneous drag force and a Brownian-type random force. More precisely, the Kolmogorov type diffusion $(V,X)$ is considered: here $X$ is the position of the particle and $V$ is its velocity and is solution of a stochastic differential equation driven by a one-dimensional Brownian motion, with the drift of the form $t^{-β}F(v)$. The function $F$ satisfies some homogeneity condition and $β$ is positive. The behaviour of the process $(V,X)$ in large time is proved by using stochastic analysis tools.
We investigate the long-time evolution of branching diffusion processes (starting with a finite number of particles) in inhomogeneous media. The qualitative behavior of the processes depends on the intensity of the branching. In the super-critical regime, we describe the asymptotics of the number of particles in a given domain. In the sub-critical and critical regimes, we show that the limiting number of particles is finite and describe its distribution.
Freidlin-Wentzell type exit-time estimates for time-inhomogeneous diffusions and their applications
本文研究非齐次扩散的逃逸时间，证明在漂移和扩散项一致接近时逃逸时间指数增长，并推广至McKean-Vlasov过程。
On weak and strong solutions of time inhomogeneous Itô's equations with VMO diffusion and Morrey drift
该研究针对带非正则扩散项与Morrey空间漂移项的非齐次伊藤随机方程，证明了弱解存在性，讨论了条件弱唯一性与强解相关猜想，结果在漂移项为零时仍具创新性。
Let us consider a solution of the time-inhomogeneous stochastic differential equation driven by a Brownian motion with drift coefficient $b(t,x)=ρ\,{\rm sgn}(x)|x|^α/t^β$. This process can be viewed as a distorted Brownian motion in a potential, possibly singular, depending on time. After obtaining results on existence and uniqueness of solution, we study its asymptotic behaviour and made a precise description, in terms of parameters $ρ,α$ and $β$, of the recurrence, transience and convergence. More precisely, asymptotic distributions, iterated logarithm type laws and rates of transience and e…
This paper studies time-inhomogeneous nonequilibrium diffusion processes, including both Brownian dynamics and Langevin dynamics. We derive upper bounds of the relative entropy production of the time-inhomogeneous process with respect to the transient invariant probability measures. We also study the time reversal of the reverse process in Crooks' fluctuation theorem. We show that the time reversal of the reverse process coincides with the optimally controlled forward process that leads to zero variance importance sampling estimator based on Jarzynski's equality.
The main result in this paper is a variational formula for the exit rate from a bounded domain for a diffusion process in terms of the stationary law of the diffusion constrained to remain in this domain forever. Related results on the geometric ergodicity of the controlled Q-process are also presented.
Let $L_t:=Δ_t+Z_t$ for a $C^{1,1}$-vector field $Z$ on a differential manifold $M$ with boundary $\partial M$, where $Δ_t$ is the Laplacian induced by a time dependent metric $g_t$ differentiable in $t\in [0,T_c)$. We first introduce the reflecting diffusion process generated by $L_t$ and establish the derivative formula for the associated diffusion semigroup; then construct the couplings for the reflecting $L_t$-diffusion processes by parallel and reflecting displacement, which implies the gradient estimates of the associated heat semigroup; and finally, present a number of equivalent inequal…
The paper considers an Euler discretization based numerical scheme for approximating functionals of invariant distribution of an ergodic diffusion. Convergence of the numerical scheme is shown for suitably chosen discretization step, and a thorough error analysis is conducted by proving central limit theorem and moderate deviation principle for the error term. The paper is a first step in understanding efficiency of discretization based numerical schemes for estimating invariant distributions, which is comparatively much less studied than the schemes used for generating approximate trajectorie…
Nonequilibrium fluctuation-dissipation theorems (FDTs) are one of the most important advances in stochastic thermodynamics over the past two decades. Here we provide rigorous mathematical proofs of two types of nonequilibrium FDTs for inhomogeneous diffusion processes with unbounded drift and diffusion coefficients by using the Schauder estimates for partial differential equations of parabolic type and the theory of weakly continuous semigroups. The FDTs proved in this paper apply to any forms of inhomogeneous and nonlinear external perturbations. Furthermore, we prove the uniqueness of the co…
We consider a mean field game with common noise in which the diffusion coefficients may be controlled. We prove existence of a weak relaxed solution under some continuity conditions on the coefficients. We then show that, when there is no common noise, the solution of this mean field game is characterized by a McKean-Vlasov type second order backward SDE.
We consider a time inhomogeneous strong Markov process $(ξ_t)_{t\ge 0}$ taking values in a Polish state space whose semigroup has a $T$-periodic structure. We give simple conditions which imply ergodicity of the grid chain $(ξ_{kT})_{k\in \mathbb{N}_0}$ In case of $d$-dimensional possibly degenerate diffusions, the conditions are stated in terms of drift and diffusion coefficient of the process.
We prove the solvability of Itô stochastic equations with uniformly nondegenerate, bounded, measurable diffusion and drift in $L_{d+1}(\mathbb{R}^{d+1})$. Actually, the powers of summability of the drift in $x$ and $t$ could be different. Our results seem to be new even if the diffusion is constant. The method of proving the solvability belongs to A.V. Skorokhod. Weak uniqueness of solutions is an open problem even if the diffusion is constant.
We prove the global asymptotic equivalence between the experiments generated by the discrete (high frequency) or continuous observation of a path of a time inhomogeneous jump-diffusion process and a Gaussian white noise experiment. Here, the considered parameter is the drift function, and we suppose that the observation time $T$ tends to $\infty$. The approximation is given in the sense of the Le Cam $Δ$-distance, under smoothness conditions on the unknown drift function. These asymptotic equivalences are established by constructing explicit Markov kernels that can be used to reproduce one exp…
Markov approximation for controlled Hawkes Jump-Diffusions with general kernels
本文提出了一种针对具有一般核的Hawkes跳扩散的Markov逼近方法，通过用指数函数线性组合逼近核，证明其可由Markov跳扩散逼近，并应用于随机控制问题验证了实用性。
An inhomogeneous controlled branching process
本文研究后代分布世代依赖、繁殖数受随机机制控制的离散时间非齐次受控分支过程，证明其满足分支过程经典二象性，给出灭绝与无限增长的充分条件，并分析了非灭绝情形下的增长速率。
We consider an elliptic and time-inhomogeneous diffusion process with time-periodic coefficients evolving in a bounded domain of $\mathbb{R}^d$ with a smooth boundary. The process is killed when it hits the boundary of the domain (hard killing) or after an exponential time (soft killing) associated with some bounded rate function. The branching particle interpretation of the non absorbed diffusion again behaves as a set of interacting particles evolving in an absorbing medium. Between absorption times, the particles evolve independently one from each other according to the diffusion semigroup;…
Comparison results are given for time-inhomogeneous Markov processes with respect to function classes induced stochastic orderings. The main result states comparison of two processes, provided that the comparability of their infinitesimal generators as well as an invariance property of one process is assumed. The corresponding proof is based on a representation result for the solutions of inhomogeneous evolution problems in Banach spaces, which extends previously known results from the literature. Based on this representation, an ordering result for Markov processes induced by bounded and unbo…
In this paper, the strong existence and uniqueness for a degenerate finite system of quantile-dependent McKean-Vlasov stochastic differential equations are obtained under a weak Hörmander condition. The approach relies on the apriori bounds for the density of the solution to time inhomogeneous diffusions. The time inhomogeneous Feynman-Fac formula is used to construct a contraction map for this degenerate system.
The aim of the paper is to address the behavior in large population of diffusions interacting on a random, possibly diluted and inhomogeneous graph. This is the natural continuation of a previous work, where the homogeneous Erd\H os-Rényi case was considered. The class of graphs we consider includes disordered $W$-random graphs, with possibly unbounded graphons. The main result concerns a quenched convergence (that is true for almost every realization of the random graph) of the empirical measure of the system towards the solution of a nonlinear Fokker-Planck PDE with spatial extension, also a…
We solve the Skorokhod embedding problem for a class of stochastic processes satisfying an inhomogeneous stochastic differential equation (SDE) of the form $d A_t =μ(t, A_t) d t + σ(t, A_t) d W_t$. We provide sufficient conditions guaranteeing that for a given probability measure $ν$ on $\mathbb{R}$ there exists a bounded stopping time $τ$ and a real $a$ such that the solution $(A_t)$ of the SDE with initial value $a$ satisfies $A_τ\sim ν$. We hereby distinguish the cases where $(A_t)$ is a solution of the SDE in a weak or strong sense. Our construction of embedding stopping times is based on …
A reverse ergodic theorem for inhomogeneous killed Markov chains and application to a new uniqueness result for reflecting diffusions
该研究证明了非齐次次概率转移函数的反向遍历定理，据此得到含单个奇点区域上斜反射半鞅扩散的存在唯一性，并将反射布朗运动的相关半鞅结果部分推广至高维。
Consider the random graph on $n$ vertices $1, ..., n$. Each vertex $i$ is assigned a type $X_i$ with $X_1, ..., X_n$ being independent identically distributed as a nonnegative discrete random variable $X$. We assume that ${\bf E} X^3<\infty$. Given types of all vertices, an edge exists between vertices $i$ and $j$ independent of anything else and with probability $\min \{1, \frac{X_iX_j}{n}(1+\frac{a}{n^{1/3}}) \}$. We study the critical phase, which is known to take place when ${\bf E} X^2=1$. We prove that normalized by $n^{-2/3}$ the asymptotic joint distributions of component sizes of the …
We study asymptotic behavior of maximum likelihood estimator for a time inhomogeneous diffusion process given by a SDE $dX_t=αb(t)X_t dt + σ(t) dB_t$, $t\in[0,T)$, with a parameter $α\in R$, where $T\in(0,\infty]$ and $(B_t)_{t\in[0,T)}$ is a standard Wiener process. We formulate sufficient conditions under which the MLE of $α$ normalized by Fisher information converges to the limit distribution of Dickey-Fuller statistics. Next we study a SDE $dY_t=αb(t)a(Y_t) dt + σ(t) dB_t$, $t\in[0,T)$, with a perturbed drift satisfying $a(x)=x+O(1+|x|^γ)$ with some $γ\in[0,1)$. We give again sufficient co…
We provide an original and general sufficient criterion ensuring the exponential contraction of Feynman-Kac semi-groups of penalized processes. This criterion is applied to time-inhomogeneous one-dimensional diffusion processes conditioned not to hit 0 and to penalized birth and death processes evolving in a quenched random environment.
We consider a time inhomogeneous jump Markov process $X = (X_t)_t$ with state dependent jump intensity, taking values in $R^d . $ Its infinitesimal generator is given by \begin{multline*} L_t f (x) = \sum_{i=1}^d \frac{\partial f}{\partial x_i } (x) b^i ( t,x) - \sum_{ i =1}^d \frac{\partial f}{\partial x_i } (x) \int_{E_1} c_1^i ( t, z, x) γ_1 ( t, z, x ) μ_1 (dz ) \\ + \sum_{l=1}^3 \int_{E_l} [ f ( x + c_l ( t, z, x)) - f(x)] γ_l ( t, z, x) μ_l (dz ) , \end{multline*} where $(E_l , {\mathcal E}_l, μ_l ) , 1 \le l \le 3, $ are sigma-finite measurable spaces describing three different jump reg…
A coupling by reflection of a time-inhomogeneous diffusion process on a manifold are studied. The condition we assume is a natural time-inhomogeneous extension of lower Ricci curvature bounds. In particular, it includes the case of backward Ricci flow. As in time-homogeneous cases, our coupling provides a gradient estimate of the diffusion semigroup which yields the strong Feller property. To construct the coupling via discrete approximation, we establish the convergence in law of geodesic random walks as well as a uniform non-explosion type estimate.
The recently introduced two-parameter Poisson-Dirichlet diffusion extends the infinitely-many-neutral-alleles model, related to Kingman's distribution and to Fleming-Viot processes. The role of the additional parameter has been shown to regulate the clustering structure of the population, but is yet to be fully understood in the way it governs the reproductive process. Here we shed some light on these dynamics by providing a finite-population construction, with finitely-many species, of the two-parameter infinite-dimensional diffusion. The costruction is obtained in terms of Wright-Fisher chai…
Existence of densities and atoms for the running maximum of time-inhomogeneous jump diffusions
该研究针对时间非齐次跳扩散过程运行最大值的分布性质，运用Malliavin分析等工具，证明两种情形下的绝对连续性，还揭示静默初始区间的原子效应并给出分解。
In this note, we propose two different approaches to rigorously justify a pseudo-Markov property for controlled diffusion processes which is often (explicitly or implicitly) used to prove the dynamic programming principle in the stochastic control literature. The first approach develops a sketch of proof proposed by Fleming and Souganidis~\cite{fleming-souganidis}. The second approach is based on an enlargement of the original state space and a controlled martingale problem. We clarify some measurability and topological issues raised by these two approaches.
In this paper, we study the asymptotic of exit problem for controlled Markov diffusion processes with random jumps and vanishing diffusion terms, where the random jumps are introduced in order to modify the evolution of the controlled diffusions by switching from one mode of dynamics to another. That is, depending on the state-position and state-transition information, the dynamics of the controlled diffusions randomly switches between the different drift and diffusion terms. Here, we specifically investigate the asymptotic exit problem concerning such controlled Markov diffusion processes in …
We consider a diffusion $(ξ_t)_{t\ge 0}$ whose drift contains some deterministic periodic signal. Its shape being fixed and known, up to scaling in time, the periodicity of the signal is the unknown parameter $\vartheta$ of interest. We consider sequences of local models at $\vartheta$, corresponding to continuous observation of the process $ξ$ on the time interval $[0,n]$ as $n\to\infty$, with suitable choice of local scale at $\vartheta$. Our tools --under an ergodicity condition-- are path segments of $ξ$ corresponding to the period $\vartheta$, and limit theorems for certain functionals of…
We consider weakly interacting diffusions on time varying random graphs. The system consists of a large number of nodes in which the state of each node is governed by a diffusion process that is influenced by the neighboring nodes. The collection of neighbors of a given node changes dynamically over time and is determined through a time evolving random graph process. A law of large numbers and a propagation of chaos result is established for a multi-type population setting where at each instant the interaction between nodes is given by an inhomogeneous random graph which may change over time. …
We consider a controlled second order differential equation which is partially observed with an additional fractional noise. we study the asymptotic (for large observation time) design problem of the input and give an efficient estimator of the unknown signal drift parameter. When the input depends on the unknow parameter, we will try the one-step estimation procedure using the Newton-Raphson method.
Let $τ= (τ_i : i \in {\Bbb Z})$ denote i.i.d.~positive random variables with common distribution $F$ and (conditional on $τ$) let $X = (X_t : t\geq0, X_0=0)$, be a continuous-time simple symmetric random walk on ${\Bbb Z}$ with inhomogeneous rates $(τ_i^{-1} : i \in {\Bbb Z})$. When $F$ is in the domain of attraction of a stable law of exponent $α<1$ (so that ${\Bbb E}(τ_i) = \infty$ and X is subdiffusive), we prove that $(X,τ)$, suitably rescaled (in space and time), converges to a natural (singular) diffusion $Z = (Z_t : t\geq0, Z_0=0)$ with a random (discrete) speed measure $ρ$. The converg…
We consider a process $(X_t)_{t\in[0,T)}$ given by the SDE $dX_t = αb(t)X_t dt + σ(t) dB_t$, $t\in[0,T)$, with initial condition $X_0=0$, where $T\in(0,\infty]$, $α\in R$, $(B_t)_{t\in[0,T)}$ is a standard Wiener process, $b:[0,T)\to R\setminus\{0\}$ and $σ:[0,T)\to(0,\infty)$ are continuously differentiable functions. Assuming that $b$ and $σ$ satisfy a certain differential equation we derive an explicit formula for the joint Laplace transform of $\int_0^t\frac{b(s)^2}{σ(s)^2}(X_s)^2 ds$ and $(X_t)^2$ for all $t\in[0,T)$. As an application, we study asymptotic behavior of the maximum likeliho…
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