arXiv subjects

Huaizhong Zhao

Publications and source records attributed to Huaizhong Zhao.

At least 19 recordsLinked to original sources

arXiv2026

Ergodic Stochastic Optimal Control Problems

In this article, we introduce a novel approach to solving the ergodic stochastic optimal control problem whose dynamics is driven by a controlled stochastic differential equation. The coefficients of this stochastic differential equation are non-autonomous but periodic in time. We first prove that the infinite horizon average stochastic optimal control problem is ergodic, i.e., the value function equals a constant $ρ$ which is independent of the initial conditions. Based on the result of ergodicity, we construct an auxiliary function $w(t,x)$ that is well-defined and periodic in time. We can prove that the pair $(w,ρ)$ satisfies the dynamic programming principle and is a viscosity solution of the associated Hamilton-Jacobi-Bellman equation. Finally, we apply our results to the study of ergodic backward stochastic differential equations. Our method is even new in the homogeneous case.

math.OC↗

arXiv2026

Strong averaging principle for multiscale time-inhomogeneous SDEs with multiplicative $α$-stable noises

In this paper, we study the strong averaging principle for multiscale time-inhomogeneous stochastic systems driven by multiplicative $α$-stable processes with $α\in(1,2)$. Based on Khasminskii's discretization approach, we first establish that the fast component processes with a frozen slow variable admits a periodic measure. We then prove the strong convergence of the slow subsystem to an averaged system that depends on the time scale $\varepsilon$. For any fixed $\varepsilon$, if the reciprocals of the two periods $τ_1$ and $\varepsilon τ_2$ are rationally linearly independent, an important consequence is that the averaged system has random quasi-periodicity. Furthermore, by applying the ergodic theorem, we prove the strong convergence of the slow subsystem to another averaged system, a time-inhomogeneous SDEs independent of the time scale $\varepsilon$. Our result is also novel even in the time-homogeneous case for a fully coupled multiscale system with multiplicative $α$-stable noises. Finally, we apply the result to a climate-weather system.

math.PR↗

arXiv2026

Large deviations for invariant measure of stochastic Allen-Cahn equation with inhomogeneous boundary conditions and multiplicative noise

We establish a small-noise large deviation principle for the family of invariant measures $\{μ_ε\}_{ε>0}$ associated with the one-dimensional stochastic Allen-Cahn equation, subject to inhomogeneous Dirichlet boundary conditions and driven by unbounded multiplicative noise. The main novelty is that the deterministic system is only weakly dissipative, while the noise coefficient is allowed to have strictly sublinear growth arbitrarily close to linear. Using L. Simon's convergence theorem, we prove that every trajectory of the corresponding noiseless equation converges, as time tends to infinity, to the unique minimiser of the Ginzburg-Landau energy functional determined by the boundary conditions. A key ingredient is an exponential estimate for the invariant measures outside bounded subsets of $W^{k^\star,p^\star}$, where $k^\star p^\star>1$ and $p^\star$ are sufficiently large; such subsets are compact in the underlying space of continuous functions. As a consequence of the large deviation principle, we show that, as $ε\to 0$, the invariant measures $μ_ε$ concentrate exponentially fast around the unique minimiser.

math.PR↗

arXiv2026

Exponential Mixing for 2D Stochastic Damped Euler Equation Driven by Bounded Noise

In this paper, we study the long-time behaviour of the two-dimensional stochastic damped Euler equation on the torus driven by bounded random forcing. Unlike stochastic Navier-Stokes or fractionally dissipative Euler equations, the model possesses no viscous regularisation, so the classical parabolic smoothing is unavailable. We prove that when the damping coefficient is sufficiently large, the associated Markov semigroup admits a unique invariant measure and converges exponentially fast to equilibrium. The key ingredient is the establishment of a global-in-time $W^{1,\infty}$ estimate for the vorticity. This estimate yields a compact absorbing mechanism in $C(\mathbb{T}^2)$, which enables us to establish the uniqueness of the invariant measure and exponential mixing. To the best of our knowledge, this is the first exponential mixing result for a genuinely inviscid stochastic Euler-type equation. Our approach demonstrates that sufficiently strong linear damping can effectively replace the compactness mechanism usually provided by viscosity and is expected to be applicable to other inviscid or weakly dissipative stochastic partial differential equations driven by bounded random forcing.

math.PR↗

arXiv2026

The existence of invariant sublinear expectations for $G$-SDEs

In this paper, we study the existence of invariant sublinear expectations of Markovian semigroups on sublinear expectation spaces. To achieve this, we establish a complete metric space of sublinear expectations, on which we extend Harris' method to the nonlinear setting on the convergence of sublinear semigroups. We then explore two cases of $G\text{-}$diffusions by studying the Lyapunov function and the local Doeblin condition. One is the $G\text{-}$Brownian motion on the unit circle which is the case studied in Feng and Zhao \cite{Zhaonon}, but with the new method. Another is the multidimensional $G\text{-}$SDEs on the whole space $\mathbb{R}^d$. We establish, for the first time in the literature, the existence of the invariant sublinear expectation for $G\text{-}$SDEs under the non-degenerate and weakly dissipative assumption. For this, we prove that for a class of $G\text{-}$SDEs, the $G\text{-}$expectation can be represented as the supremum of the semigroup of a family of SDEs, of which the regularity is obtained by considering the Bismut-Elworthy-Li formula and the Denis-Hu-Peng representation for the distribution of $G\text{-}$Brownian motions.

math.PR↗

arXiv2026

Entrance measures and dynamics for time-inhomogeneous McKean-Vlasov stochastic differential equations

In this paper, we study the entrance measures of time-inhomogeneous McKean-Vlasov SDEs. The existence is obtained in great generality, where the system can be expanding globally and/or degenerate for numerous number of time intervals. When the parameters are periodic/quasi-periodic in time, we obtain the existence of periodic/asymptotic quasi-periodic measures. In this case, a double-lift of the random dynamical system first to a dynamical system on cylinder and then on the graph of reparameterized process living on the cylinder is introduced. The double-lifted system gives to a continuous dynamical system over probability measures on the cylinder, and the lifted multi-parameter measure of the asymptotic quasi-periodic measure can then lead to an invariant measure of the lifted semigroup.

math.PR↗

arXiv2026

Invariant and periodic measures in classical spin systems on infinite lattices with highly degenerate noise

In this paper, we consider the classical spin systems on unbounded lattices given by infinite-dimensional stochastic differential equations (SDEs). We assume that the stochastic forcing acts only on one particle. The other particles are not subject to stochastic forcing directly, but interact with their nearest neighbouring particles. Under the above highly degenerate noise setting, with some mild assumptions on the local interaction of each particle such as weak dissipation, we obtain the existence, uniqueness and the Markovian property of weak martingale solutions. We prove that the one-dimensional noise can propagate to any spin particle in the system in the sense that there exists a unique invariant/periodic measure and geometric ergodicity holds for the Markovian system when restricted to any finite volume. We then prove the finite-dimensional invariant measure and the average of lifted periodic measure are tight, and weak convergent subsequence gives an invariant and periodic measures of the infinite spin systems, respectively, in the time-homogeneous or time-periodic cases.

math.PR↗

arXiv2026

Value existence for zero-sum ergodic stochastic differential games

In this paper we investigate two-player zero-sum stochastic differential games with an ergodic payoff, in which the diffusion coefficient does not need to be non-degenerate. We first establish the existence of a viscosity solution to the associated ergodic Hamilton-Jacobi-Bellman-Isaacs equation under a dissipativity condition. With the help of this viscosity solution, we then derive estimates for the upper and the lower ergodic value functions by constructing a series of non-degenerate approximating processes combined with the sup- and inf-convolution techniques. Finally, we prove the existence of a value for the game under the Isaacs condition and provide its representation formulae. As an application, we study the pollution accumulation problem with a long-run average social welfare to illustrate our theoretical results.

math.OC↗

arXiv2024

Large deviations principle for invariant measures of stochastic Burgers equations

We study the small noise asymptotic for stochastic Burgers equations on $(0,1)$ with Dirichlet boundary condition. We consider the case that the noise is more singular than space-time white noise. We let the noise magnitude $\sqrtε \rightarrow 0$ and the covariance operator $Q_ε$ is convergent to $(-Δ)^{\frac 1 2}$ and prove a large deviations principle for solutions, uniformly with respect to the initial value of equation. Furthermore, we set $Q_ε$ to be a trace class operator and converge to $(-Δ)^{\fracα{2}}$ with $α<1$ in a suitable way such that the invariant measures exist. Then, we prove the large deviations principle for the invariant measures of stochastic Burgers equations.

math.PR↗

arXiv2024

Finite ergodic components for upper probabilities

Under the notion of ergodicity of upper probability in the sense of Feng and Zhao (2021) that any invariant set either has capacity $0$ or its complement has capacity 0, we introduce the definition of finite ergodic components (FEC). We prove an invariant upper probability has FEC if and only if it is in the regime that any invariant set has either capacity $0$ or capacity $1$, proposed by Cerreia-Vioglio, Maccheroni, and Marinacci (2016). Furthermore, this is also equivalent to that the eigenvalue $1$ of the Koopman operator is of finite multiplicity, while in the ergodic upper probability regime, as in the classical ergodic probability case, the eigenvalue $1$ of the Koopman operator is simple. Additionally, we obtain the equivalence of the law of large numbers with multiple values, the asymptotic independence and the FEC. Furthermore, we apply these to obtain the corresponding results for non-invariant probabilities.

math.PR↗

arXiv2024

Ergodicity and Mixing of invariant capacities and applications

We introduce the notion of common conditional expectation to investigate Birkhoff's ergodic theorem and subadditive ergodic theorem for invariant upper probabilities. If in addition, the upper probability is ergodic, we construct an invariant probability to characterize the limit of the ergodic mean. Moreover, this skeleton probability is the unique ergodic probability in the core of the upper probability, that is equal to all probabilities in the core on all invariant sets. We have the following applications of these two theorems: $\bullet$ provide a strong law of large numbers for ergodic stationary sequence on upper probability spaces; $\bullet$ prove the multiplicative ergodic theorem on upper probability spaces; $\bullet$ establish a criterion for the ergodicity of upper probabilities in terms of independence. Furthermore, we introduce and study weak mixing for capacity preserving systems. Using the skeleton idea, we also provide several characterizations of weak mixing for invariant upper probabilities. Finally, we provide examples of ergodic and weakly mixing capacity preserving systems. As applications, we obtain new results in the classical ergodic theory. e.g. in characterizing dynamical properties on measure preserving systems, such as weak mixing, periodicity. Moreover, we use our results in the nonlinear theory to obtain the asymptotic independence, Birkhoff's type ergodic theorem, subadditive ergodic theorem, and multiplicative ergodic theorem for non-invariant probabilities.

math.PR↗

arXiv2023

Entrance measures for semigroups of time-inhomogeneous SDEs: possibly degenerate and expanding

In this article, we solve the problem of the long time behaviour of transition probabilities of time-inhomogeneous Markov processes and give a unified approach to stochastic differential equations (SDEs) with periodic, quasi-periodic, almost-periodic forcing and much beyond. We extend Harris's ``small set'' method to the time-inhomogeneous situation with the help of Hairer-Mattingly's refinement of Harris's recurrence to a one-step contraction of probability measures under the total variation distance $ρ_β$ weighted by some appropriate Lyapunov function and a constant $β>0$. We show that the convergence to an entrance measure under $ρ_β$ implies the convergence both in the total variation distance and the Wasserstein distance $\mathcal{W}_1$. For SDEs with locally Lipschitz and polynomial growth coefficients, in order to establish the local Doeblin condition, we obtain a nontrivial lower bound estimates for the fundamental solution of the corresponding Fokker-Planck equation. The drift term is allowed to be possibly non-weakly-dissipative, and the diffusion term can be degenerate over infinitely many large time intervals. This causes the system to be expanding over many periods of large time durations, the convergence to the entrance measure is generally only subgeometric. As an application we obtain the existence and uniqueness of quasi-periodic measure. We then lift the quasi-periodic Markovian semigroup to a cylinder on a torus and obtain a unique invariant measure and its ergodicity.

math.PR↗

arXiv2021

Ergodic Numerical Approximation to Periodic Measures of Stochastic Differential Equations

In this paper, we consider numerical approximation to periodic measure of a time periodic stochastic differential equations (SDEs) under weakly dissipative condition. For this we first study the existence of the periodic measure $ρ_t$ and the large time behaviour of $\mathcal{U}(t+s,s,x) := \mathbb{E}ϕ(X_{t}^{s,x})-\intϕdρ_t,$ where $X_t^{s,x}$ is the solution of the SDEs and $ϕ$ is a test function being smooth and of polynomial growth at infinity. We prove $\mathcal{U}$ and all its spatial derivatives decay to 0 with exponential rate on time $t$ in the sense of average on initial time $s$. We also prove the existence and the geometric ergodicity of the periodic measure of the discretized semi-flow from the Euler-Maruyama scheme and moment estimate of any order when the time step is sufficiently small (uniform for all orders). We thereafter obtain that the weak error for the numerical scheme of infinite horizon is of the order $1$ in terms of the time step. We prove that the choice of step size can be uniform for all test functions $ϕ$. Subsequently we are able to estimate the average periodic measure with ergodic numerical schemes.

math.PR↗

arXiv2021

Ergodicity of Sublinear Markovian Semigroups

In this paper, we study the ergodicity of invariant sublinear expectation of sublinear Markovian semigroup. For this, we first develop an ergodic theory of an expectation-preserving map on a sublinear expectation space. Ergodicity is defined as any invariant set either has $0$ capacity itself or its complement has $0$ capacity. We prove, under a general sublinear expectation space setting, the equivalent relation between ergodicity and the corresponding transformation operator having simple eigenvalue $1$, and also with Birkhoff type strong law of large numbers if the sublinear expectation is regular. For sublinear Markov process, we prove that its ergodicity is equivalent to the Markovian semigroup having eigenvalue $1$ and it is simple in the space of bounded measurable functions. As an example we show that $G$-Brownian motion $\{B_t\}_{t\geq 0}$ on the unit circle has an invariant expectation and is ergodic if and only if ${\mathbb E}(-(B_1)^2)<0$. Moreover, it is also proved in this case that the invariant expectation is regular and the canonical stationary process has no mean-uncertainty under the invariant expectation.

math.PR↗

arXiv2021

Random quasi-periodic paths and quasi-periodic measures of stochastic differential equations

In this paper, we define random quasi-periodic paths for random dynamical systems and quasi-periodic measures for Markovian semigroups. We give a sufficient condition for the existence and uniqueness of random quasi-periodic paths and quasi-periodic measures for stochastic differential equations and a sufficient condition for the density of the quasi-periodic measure to exist and to satisfy the Fokker-Planck equation. We obtain an invariant measure by considering lifted flow and semigroup on cylinder and the tightness of the average of lifted quasi-periodic measures. We further prove that the invariant measure is unique, and thus ergodic.

math.PR↗

arXiv2021

Expected Exit Time for Time-Periodic Stochastic Differential Equations and Applications to Stochastic Resonance

In this paper, we derive a parabolic partial differential equation for the expected exit time of non-autonomous time-periodic non-degenerate stochastic differential equations. This establishes a Feynman-Kac duality between expected exit time of time-periodic stochastic differential equations and time-periodic solutions of parabolic partial differential equations. Casting the time-periodic solution of the parabolic partial differential equation as a fixed point problem and a convex optimisationproblem, we give sufficient conditions in which the partial differential equation is well-posed in a weak and classical sense. With no known closed formulae for the expected exit time, we show our method can be readily implemented by standard numerical schemes. With relatively weak conditions (e.g. locally Lipschitz coefficients), the method in this paper is applicable to wide range of physical systems including weakly dissipative systems. Particular applications towards stochastic resonance will be discussed.

math.PR↗

arXiv2021

A Sufficient and Necessary Condition of PS-ergodicity of Periodic Measures and Generated Ergodic Upper Expectations

This paper contains two parts. In the first part, we study the ergodicity of periodic measures of random dynamical systems on a separable Banach space. We obtain that the periodic measure of the continuous time skew-product dynamical system generated by a random periodic path is ergodic if and only if the underlying noise metric dynamical system at discrete time of integral multiples of the period is ergodic. For the Markov random dynamical system case, we prove that the periodic measure of a Markov semigroup is PS-ergodic if and only if the trace of the random periodic path at integral multiples of period either entirely lies on a Poincaré section or completely outside a Poincaré section almost surely. In the second part of this paper, we construct sublinear expectations from periodic measures and obtain the ergodicity of the sublinear expectations from the ergodicity of periodic measures. We give some examples including the ergodicity of the discrete time Wiener shift of Brownian motions. The latter result would have some independent interests.

math.PR↗

arXiv2020

Random Periodic Processes, Periodic Measures and Ergodicity

Ergodicity of random dynamical systems with a periodic measure is obtained on a Polish space. In the Markovian case, the idea of Poincaré sections is introduced. It is proved that if the periodic measure is PS-ergodic, then it is ergodic. Moreover, if the infinitesimal generator of the Markov semigroup only has equally placed simple eigenvalues including $0$ on the imaginary axis, then the periodic measure is PS-ergodic and has positive minimum period. Conversely if the periodic measure with the positive minimum period is PS-mixing, then the infinitesimal generator only has equally placed simple eigenvalues (infinitely many) including $0$ on the imaginary axis. Moreover, under the spectral gap condition, PS-mixing of the periodic measure is proved. The ``equivalence" of random periodic processes and periodic measures is established. This is a new class of ergodic random processes. Random periodic paths of stochastic perturbation of the periodic motion of an ODE is obtained.

math.PR↗