arXiv subjects

Huaizhong Zhao

Publications and source records attributed to Huaizhong Zhao.

At least 37 records · Page 2Linked to original sources

arXiv2019

Existence of Geometric Ergodic Periodic Measures of Stochastic Differential Equations

Periodic measures are the time-periodic counterpart to invariant measures for dynamical systems and can be used to characterise the long-term periodic behaviour of stochastic systems. This paper gives sufficient conditions for the existence, uniqueness and geometric convergence of a periodic measure for time-periodic Markovian processes on a locally compact metric space in great generality. In particular, we apply these results in the context of time-periodic weakly dissipative stochastic differential equations, gradient stochastic differential equations as well as Langevin equations. We will establish the Fokker-Planck equation that the density of the periodic measure sufficiently and necessarily satisfies. Applications to physical problems shall be discussed with specific examples.

math.PR↗

arXiv2018

Random periodic solutions and ergodicity for stochastic differential equations

In this paper, we establish some sufficient conditions for the existence of stable random periodic solutions of stochastic differential equations and ergodicity in the random periodic regime. The techniques involve the existence of Lyapunov type function, using two-point generator of the stochastic flow map, strong Feller argument and weak convergence.

math.PR↗

arXiv2018

Ergodicity of Invariant Capacity

In this paper, we investigate capacity preserving transformations and their ergodicity. We show that for any measurable transformation $θ$ there always exists a $θ$-invariant capacity. We investigate some limit properties under capacity spaces and then give the concept of ergodicity for a capacity preserving transformation. Based on this definition, we give several characterizations of ergodicity. In particular, we obtain a type of Birkhoff's ergodic theorem and prove that the ergodicity of $θ$ with respect to an upper probability is equivalent to the strong law of large numbers.

math.PR↗

arXiv2018

Anticipating Random Periodic Solutions--II. SPDEs with Multiplicative Linear Noise

In this paper, we study the existence of random periodic solutions for semilinear stochastic partial differential equations with multiplicative linear noise on a bounded open domain ${\cal O}\subset {\mathbb R}^d$ with smooth boundary. We identify them with the solutions of coupled forward-backward infinite horizon stochastic integral equations in $L^2({\cal O})$. We then use generalized Schauder's fixed point theorem, the relative compactness of Wiener-Sobolev spaces in $C^0([0, T], L^2(Ω\times{\cal O}))$ and a localization argument to prove the existence of solutions of the infinite horizon integral equations, which immediately implies the existence of the random periodic solution to the corresponding SPDEs. As an example, we apply our result to the stochastic Allen-Cahn equation with a periodic potential and prove the existence of a random periodic solution using a localisation argument.

math.PR↗

arXiv2017

Numerical Approximation of Random Periodic Solutions of Stochastic Differential Equations

In this paper, we discuss the numerical approximation of random periodic solutions (r.p.s.) of stochastic differential equations (SDEs) with multiplicative noise. We prove the existence of the random periodic solution as the limit of the pull-back flow when the starting time tends to $-\infty$ along the multiple integrals of the period. As the random periodic solution is not explicitly constructible, it is useful to study the numerical approximation. We discretise the SDE using the Euler-Maruyama scheme and moldiflied Milstein scheme. Subsequently we obtain the existence of the random periodic solution as the limit of the pull-back of the discretised SDE. We prove that the latter is an approximated random periodic solution with an error to the exact one at the rate of $\sqrt {Δt}$ in the mean-square sense in Euler-Maruyama method and $Δt$ in the Milstein method. We also obtain the weak convergence result for the approximation of the periodic measure.

math.NA↗

arXiv2017

Rough path properties for local time of symmetric $α$ stable process

In this paper, we first prove that the local time associated with symmetric $α$-stable processes is of bounded $p$-variation for any $p>\frac{2}{α-1}$ partly based on Barlow's estimation of the modulus of the local time of such processes.\,\,The fact that the local time is of bounded $p$-variation for any $p>\frac{2}{α-1}$ enables us to define the integral of the local time $\int_{-\infty}^{\infty}\triangledown_-^{α-1}f(x)d_x L_t^x$ as a Young integral for less smooth functions being of bounded $q$-varition with $1\leq q<\frac{2}{3-α}$. When $q\geq \frac{2}{3-α}$, Young's integration theory is no longer applicable. However, rough path theory is useful in this case. The main purpose of this paper is to establish a rough path theory for the integration with respect to the local times of symmetric $α$-stable processes for $\frac{2}{3-α}\leq q< 4$.

math.PR↗

arXiv2016

Anticipating Random Periodic Solutions--I. SDEs with Multiplicative Linear Noise

In this paper, we study the existence of random periodic solutions for semilinear stochastic differential equations. We identify them as solutions of coupled forward-backward infinite horizon stochastic integral equations (IHSIEs), using the "substitution theorem" of stochastic differential equations with anticipating initial conditions. In general, random periodic solutions and the solutions of IHSIEs, are anticipating. For the linear noise case, with the help of the exponential dichotomy given in the multiplicative ergodic theorem, we can identify them as the solutions of infinite horizon random integral equations (IHSIEs). We then solve a localised forward-backward IHRIE in $C(\mathbb{R}, L^2_{loc}(Ω))$ using an argument of truncations, the Malliavin calculus, the relative compactness of Wiener-Sobolev spaces in $C([0, T], L^2(Ω))$ and Schauder's fixed point theorem. We finally measurably glue the local solutions together to obtain a global solution in $C(\mathbb{R}, L^2(Ω))$. Thus we obtain the existence of a random periodic solution and a periodic measure.

math.PR↗

arXiv2015

Random Periodic Solutions of Random Dynamical Systems

In this paper, we give the definition of the random periodic solutions of random dynamical systems. We prove the existence of such periodic solutions for a $C^1$ perfect cocycle on a cylinder using a random invariant set, the Lyapunov exponents and the pullback of the cocycle.

math.PR↗

arXiv2015

Pathwise Random Periodic Solutions of Stochastic Differential Equations

In this paper, we study the existence of random periodic solutions for semilinear stochastic differential equations. We identify these as the solutions of coupled forward-backward infinite horizon stochastic integral equations in general cases. We then use the argument of the relative compactness of Wiener-Sobolev spaces in $C^0([0, T], L^2(Ω))$ and generalized Schauder's fixed point theorem to prove the existence of a solution of the coupled stochastic forward-backward infinite horizon integral equations. The condition on $F$ is then further weakened by applying the coupling method of forward and backward Gronwall inequalities. The results are also valid for stationary solutions as a special case when the period $τ$ can be an arbitrary number.\

math.PR↗

arXiv2015

Random Periodic Solutions of SPDEs via Integral Equations and Wiener-Sobolev Compact Embedding

In this paper, we study the existence of random periodic solutions for semilinear SPDEs on a bounded domain with a smooth boundary. We identify them as the solutions of coupled forward-backward infinite horizon stochastic integral equations on $L^2(D)$ in general cases. For this we use Mercer's Theorem and eigenvalues and eigenfunctions of the second order differential operators in the infinite horizon integral equations. We then use the argument of the relative compactness of Wiener-Sobolev spaces in $C^0([0, T], L^2(Ω\times D))$ and generalized Schauder's fixed point theorem to prove the existence of a solution of the integral equations. This is the first paper in literature to study random periodic solutions of SPDEs. Our result is also new in finding semi-stable stationary solution for non-dissipative SPDEs, while in literature the classical method is to use the pull-back technique so researchers were only able to find stable stationary solutions for dissipative systems.

math.PR↗

arXiv2011

SPDEs with Polynomial Growth Coefficients and Malliavin Calculus Method

In this paper we study the existence and uniqueness of the $L_ρ^{2p}(\mathbb{R}^d;\mathbb{R}^1)\times L_ρ^2(\mathbb{R}^d;\mathbb{R}^d)$ valued solution of backward doubly stochastic differential equations with polynomial growth coefficients using week convergence, equivalence of norm principle and Wiener-Sobolev compactness arguments. Then we establish a new probabilistic representation of the weak solutions of SPDEs with polynomial growth coefficients through the solutions of the corresponding backward doubly stochastic differential equations (BDSDEs). This probabilistic representation is then used to prove the existence of stationary solution of SPDEs on $\mathbb{R}^d$ via infinite horizon BDSDEs. The convergence of the solution of BDSDE to the solution of infinite horizon BDSDEs is shown to be equivalent to the convergence of the pull-back of the solutions of SPDEs. With this we obtain the stability of the stationary solutions as well.

math.PR↗

arXiv2011

Probabilistic Representation of Weak Solutions of Partial Differential Equations with Polynomial Growth Coefficients

In this paper we develop a new weak convergence and compact embedding method to study the existence and uniqueness of the $L_ρ^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes L_ρ^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solution of backward stochastic differential equations with p-growth coefficients. Then we establish the probabilistic representation of the weak solution of PDEs with p-growth coefficients via corresponding BSDEs.

math.PR↗

arXiv2009

Variation and Rough Path Properties of Local Times of Lévy Processes

In this paper, we will prove that the local time of a Lévy process is of finite $p$-variation in the space variable in the classical sense, a.s. for any $p>2$, $t\geq 0$, if the Lévy measure satisfies $\int_{R\setminus \{0\}}(|y|^{3\over 2}\wedge 1)n(dy)<\infty$, and is a rough path of roughness $p$ a.s. for any $2<p<3$ under a slightly stronger condition for the Lévy measure. Then for any function $g$ of finite $q$-variation ($1\leq q <3$), we establish the integral $\int_{-\infty}^{\infty}g(x)dL_t^x$ as a Young integral when $1\leq q<2$ and a Lyons' rough path integral when $2\leq q<3$. We therefore apply these path integrals to extend the Tanaka-Meyer formula for a continuous function $f$ if $\nabla ^-f$ exists and is of finite $q$-variation when $1\leq q<3$, for both continuous semi-martingales and a class of Lévy processes.

math.PR↗

arXiv2008

Stationary Solutions of SPDEs and Infinite Horizon BDSDEs

In this paper we study the existence of stationary solutions for stochastic partial differential equations. We establish a new connection between $L_ρ^2({\mathbb{R}^{d}};{\mathbb{R}^{1}}) \otimes L_ρ^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solutions of backward doubly stochastic differential equations (BDSDEs) on infinite horizon and the stationary solutions of the SPDEs. Moreover, we prove the existence and uniqueness of the solutions of BDSDEs on both finite and infinite horizons, so obtain the solutions of initial value problems and the stationary solutions (independent of any initial value) of SPDEs. The connection of the weak solutions of SPDEs and BDSDEs has independent interests in the areas of both SPDEs and BSDEs.

math.PR↗

arXiv2008

Stationary Solutions of SPDEs and Infinite Horizon BDSDEs with Non-Lipschitz Coefficients

We prove a general theorem that the $L_ρ^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes L_ρ^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solution of an infinite horizon backward doubly stochastic differential equation, if exists, gives the stationary solution of the corresponding stochastic partial differential equation. We prove the existence and uniqueness of the $L_ρ^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes L_ρ^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solutions for backward doubly stochastic differential equations on finite and infinite horizon with linear growth without assuming Lipschitz conditions, but under the monotonicity condition. Therefore the solution of finite horizon problem gives the solution of the initial value problem of the corresponding stochastic partial differential equations, and the solution of the infinite horizon problem gives the stationary solution of the SPDEs according to our general result.

math.PR↗

arXiv2006

Pathwise stationary solutions of stochastic Burgers equations with $L^2[0,1]$-noise and stochastic Burgers integral equations on infinite horizon

In this paper, we show the existence and uniqueness of the stationary solution $u(t,ω)$ and stationary point $Y(ω)$ of the differentiable random dynamical system $U:R\times L^2[0,1]\times Ω\to L^2[0,1]$ generated by the stochastic Burgers equation with $L^2[0,1]$-noise and large viscosity, especially, $u(t,ω)=U(t,Y(ω),ω)=Y(θ(t,ω))$, and $Y(ω) \in H^1[0,1]$ is the unique solution of the following equation in $L^2[0,1]$ $$ Y(ω)={1/2}\int_{-\infty}^0T_ν(-s)\frac{\partial (Y(θ(s,ω))^2}{\partial x}ds +\int_{-\infty}^0T_ν(-s)dW_s(ω), $$ where $θ$ is the group of $P$-preserving ergodic transformation on the canonical probability pace $(Ω, {\cal F}, P)$ such that $θ(t,ω)(s)=W(t+s)-W(t)$.

math.PR↗

arXiv2006

Two-parameter $p, q$-variation Paths and Integrations of Local Times

In this paper, we prove two main results. The first one is to give a new condition for the existence of two-parameter $p, q$-variation path integrals. Our condition of locally bounded $p,q$-variation is more natural and easy to verify than those of Young. This result can be easily generalized to multi-parameter case. The second result is to define the integral of local time $\int_{-\infty}^\infty\int_0^t g(s,x)d_{s,x}L_s(x)$ pathwise and then give generalized It$\hat {\rm o}$'s formula when $\nabla^-f(s,x)$ is only of bounded $p,q$-variation in $(s,x)$. In the case that $g(s,x)=\nabla^-f(s,x)$ is of locally bounded variation in $(s,x)$, the integral $\int_{-\infty}^\infty\int_0^t \nabla^-f(s,x)d_{s,x}L_s(x)$ is the Lebesgue-Stieltjes integral and was used in Elworthy, Truman and Zhao \cite{Zhao}. When $g(s,x)=\nabla^-f(s,x)$ is of only locally $p, q$-variation, where $p\geq 1$,$q\geq 1$, and $2q+1>2pq$, the integral is a two-parameter Young integral of $p,q$-variation rather than a Lebesgue-Stieltjes integral. In the special case that $f(s,x)=f(x)$ is independent of $s$, we give a new condition for Meyer's formula and $\int_{-\infty}^\infty L_t(x)d_x\nabla^-f(x)$ is defined pathwise as a Young integral. For this we prove the local time $L_t(x)$ is of $p$-variation in $x$ for each $t\geq 0$, for each $p>2$ almost surely ($p$-variation in the sense of Lyons and Young, i.e. $\sup\limits_{E: \ a finite partition of [-N,N]} \sum\limits_{i=1}^m|L_t(x_i)-L_t(x_{i-1})|^p<\infty$).

math.PR↗

arXiv2005

A Generalized It$\hat {\rm o}$'s Formula in Two-Dimensions and Stochastic Lebesgue-Stieltjes Integrals

A generalized It${\hat {\rm o}}$ formula for time dependent functions of two-dimensional continuous semi-martingales is proved. The formula uses the local time of each coordinate process of the semi-martingale, left space and time first derivatives and second derivative $\nabla_1^- \nabla_2^-f$ only which are assumed to be of locally bounded variation in certain variables, and stochastic Lebesgue-Stieltjes integrals of two parameters.The two-parameter integral is defined as a natural generalization of the It${\hat {\rm o}}$ integral and Lebesgue-Stieltjes integral through a type of It${\hat {\rm o}}$ isometry formula.

math.PR↗