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Yaozhong Hu

Publications and source records attributed to Yaozhong Hu.

At least 19 recordsLinked to original sources
arXiv2026

Density convergence of spatial average of solution to a one dimensional stochastic wave equation

In this paper we study the spatial averages of the solution of a one-dimensional stochastic wave equation driven by a Gaussian multiplicative noise, which is white in time and has a homogeneous spatial covariance described by the Riesz kernel. We establish the rate of convergence for the uniform distance between the density of spatial averages and the standard normal density. The proof combines Malliavin calculus with Stein's method for normal approximations. The key technical challenges lie in estimating the Lp-norm of the second Malliavin derivative and the existence of negative moments of Malliavin covariance matrix.

math.PR↗
arXiv2026

Temporal quartic variation for non-linear stochastic heat equations with piecewise constant coefficients

We consider a stochastic partial differential equation with piecewise constant coefficients driven by a multiplicative space-time white noise. The existence and uniqueness of the mild solution in Walsh sense is established. We mainly study the limit behavior of the temporal quartic variation of the mild solution. As an application, we deduce a consistent estimator based on corresponding results.

math.PR↗
arXiv2026

Gaussian fluctuation for spatial average of the space--time fractional stochastic heat equation

We study spatial averages of the mild solution to a one-dimensional space--time fractional stochastic heat equation driven by space--time white noise. For fixed \(t>0\), we prove a quantitative central limit theorem for the normalized spatial average over \([-R,R]\): as \(R\to\infty\), its law converges to the standard normal law at rate \(R^{-1/2}\) in total variation distance. The proof relies on the Malliavin--Stein method, combined with precise estimates for the space--time fractional heat kernel and for the Malliavin derivative of the mild solution. We further establish a functional central limit theorem.

math.PR↗
arXiv2026

A linear-quadratic partially observed Stackelberg stochastic differential game with multiple followers and its application to multi-agent formation control

In this paper, we study a linear-quadratic partially observed Stackelberg stochastic differential game problem in which a single leader and multiple followers are involved. We consider more practical formulation for partial information that none of them can observed the complete information and the followers know more than the leader. Some completely different methods including a novel state decomposition and orthogonal decomposition are applied to overcome the difficulties caused by partially observability which improves the tools and relaxes the constraint condition imposed on admissible control in the existing literature. More precisely, the followers encounter the standard linear-quadratic partially observed optimal control problems, however, a kind of forward-backward indefinite linear-quadratic partially observed optimal control problem is considered by the leader. Instead of maximum principle of forward-backward control systems, inspired by the existing work related to definite case and classical forward control system, some distinct forward-backward linear-quadratic decoupling techniques including the method of completion of squares are applied to solve the leader's problem. More interestingly, we develop the deterministic formation control in multi-agent system with a framework of Stackelberg differential game and extend it to the stochastic case. The optimal strategies are obtained by our theoretical result suitably.

math.OC↗
arXiv2025

Stochastic wave equation with additive fractional noise: solvability and global Hölder continuity

We determine the range of Hurst parameters that provide the necessary and sufficient conditions for the solvability, in $L^2(Ω)$, of the stochastic wave equation: $ \frac{\partial^2 }{\partial t^2}u(t,x) =Δu(t,x)+\dot{W}(t,x)$, where $\{ W(t,x),\ t\geq 0, x\in \mathbb{R}^d\} $ is a fractional Brownian field with temporal Hurst parameter $H_0\in[\tfrac12,1]$ and spatial Hurst parameters $H_i\in(0,1)$ for $i=1,\cdots,d$. {In particular, the solvability condition exhibits a phase transition at $H_0 = 1$.} We also obtain the sharp growth rate and the sharp Hölder continuity of the solution on the real line in the case $H_0=1/2$.

math.PR↗
arXiv2025

Non-central limit of densities of some functionals of Gaussian processes

We establish the convergence of the densities of a sequence of nonlinear functionals of an underlying Gaussian process to the density of a Gamma distribution. The key idea of our work is a new density formula for random variables in the setting of Markov diffusion generators, which yields a special representation for the density of a Gamma distribution. Via this representation, we are able to provide precise estimates on the distance between densities while developing the techniques of Malliavin calculus and Stein's method suitable to Gamma approximation at the density level. We first focus our study on the case of random variables living in a fixed Wiener chaos of an even order for which the bound for the difference of the densities can be dominated by a linear combination of moments up to order four. We then study the case of general Gaussian functionals with possibly infinite chaos expansion. Finally, we provide an application to random variables living in the second Wiener chaos.

math.PR↗
arXiv2025

Density convergence on Markov diffusion chaos via Stein's method

We study the difference between the probability density of a random variable $F$ on Markov diffusion chaos and the probability density of a general target distribution $Z$. In the special case where $F$ is a chaotic random variables and $Z$ is a Pearson target, we extend our study to the $k$-th derivatives of the densities for all $k\in \mathbb{N}$. In particular, we obtain four moment theorems for the convergence of the $k$-th derivatives of the densities of $F$ to the corresponding $k$-th derivatives of the density of a Pearson target. Our work therefore significantly extends earlier works [HLN14,BDH24] which studies density convergence of random variables on Wiener chaos to respectively the normal and Gamma targets. We provide two applications of our results. The first application is about weighted sum of i.i.d. Gamma distribution where we show convergence in laws of this weighted sum to another Gamma distribution automatically implies convergence in densities. In the second application, we show that for a large class of Pearson diffusions, the density of its solution with any initial condition exponentially converge to its limiting density. Moreover, this exponential convergence holds for the $k$-th derivatives of the densities for all $k\in \mathbb{N}$.

math.PR↗
arXiv2025

Long time strong convergence analysis of one-step methods for McKean-Vlasov SDEs with superlinear growth coefficients

This paper presents a strong convergence rate analysis of general discretization approximations for McKean-Vlasov SDEs with super-linear growth coefficients over infinite time horizon. Under some specified non-globally Lipschitz conditions, we derive the propagation of chaos, and the mean-square convergence rate over infinite time horizon for general one-step time discretization schemes for the underlying Mckean-Vlasov SDEs. As an application of the general result it is obtained the mean-square convergence rate over infinite time horizon for two numerical schemes: the projected Euler scheme and the backward Euler scheme for Mckean-Vlasov SDEs in non-globally Lipschitz settings. Numerical experiments are provided to validate the theoretical findings.

math.NA↗
arXiv2025

Approximation of Elliptic Equations with Interior Single-Point Degeneracy and Its Application to Weak Unique Continuation Property

This paper investigates the quantitative weak unique continuation property (QWUCP) for a class of high-dimensional elliptic equations with interior point degeneracy. First, we establish well-posedness results in weighted function spaces. Then, using an innovative approximation method, we derive the three-ball theorem at the degenerate point. Finally, we apply the three-ball theorem to prove QWUCP for two different cases.

math.AP↗
arXiv2025

Feynman-Kac formula gor general time dependent stochastic parabolic equation on a bounded domain and applications

This paper establishes a Feynman-Kac formula to represent the solution to general time inhomogeneous stochastic parabolic partial differential equations driven by multiplicative fractional Gaussian noises in bounded domain where L_t is a second order uniformly elliptic operator whose coefficients can depend on time and generates a time inhomoegenous Markov process. The idea is to use the Aronson bounds of fundamental solution of the associated heat kernel and the techniques from Malliavin calculus. The newly obtained Feynman-Kac formula is then applied to establish the Holder regularity in the space and time variables. The dependence on time of the coefficients poses serious challenges and new results about the stochastic differential equations are discovered to face the challenge. An amazing application of the Feynman-Kac formula is about the matching upper and lower bounds for all moments of the solution, an critical tool for the intermittency. For the latter result we need first to establish new small ball like bounds for the diffusion associated with the parabolic differential operator in the bounded domain which is of interest on its own.

math.PR↗
arXiv2024

Limit error distributions of Milstein scheme for stochastic Volterra equations with singular kernels

For stochastic Volterra equations driven by standard Brownian and with singular kernels $K(u)=u^{H-\frac{1}{2}}/Γ(H+1/2), H\in (0,1/2)$, it is known that the Milstein scheme has a convergence rate of $n^{-2H}$. In this paper, we show that this rate is optimal. Moreover, we show that the error normalized by $n^{-2H}$ converge stably in law to the (nonzero) solution of a certain linear Volterra equation of random coefficients with the same fractional kernel.

math.PR↗
arXiv2024

Asymptotic behaviors for Volterra type McKean-Vlasov stochastic integral equations with small noise

This work is devoted to studying asymptotic behaviors for Volterra type McKean-Vlasov stochastic differential equations with small noise. By applying the weak convergence approach, we establish the large and moderate deviation principles. In addition, we obtain the central limit theorem and find the Volterra integral equation satisfied by the limiting process, which involves the Lions derivative of the drift coefficient.

math.PR↗
arXiv2024

In search of necessary and sufficient conditions to solve parabolic Anderson model with rough noise

This paper attempts to obtain necessary and sufficient conditions to solve the parabolic Anderson model with fractional Gaussian noises: $\frac{\partial}{\partial t}u(t,x)=\frac{1}{2}Δu(t,x)+u(t,x)\dot{W}(t,x)$, where $ {W}(t,x)$ is the fractional Brownian field with temporal Hurst parameter $H_0\in [1/2, 1) $ and spatial Hurst parameters $H$ $ =(H_1, \cdots, H_d)$ $ \in (0, 1)^d$, and $\dot{W}(t,x)=\frac{\partial ^{d+1}}{\partial t \partial x_1 \cdots \partial x_d}W(t,x)$. When $d=1$ and when $(H_0,H)\in(\frac 12,1)\times(\frac 1{20},\frac 12)$ we show that the condition $2H_0+H>5/2$ is necessary and sufficient to ensure the existence of a unique solution for the parabolic Anderson Model. When $d\ge 2$, we find the necessary and sufficient condition on the Hurst parameters so that each chaos of the solution candidate is square integrable.

math.PR↗
arXiv2024

Strong solution of stochastic differential equations with discontinuous and unbounded coefficients

In this paper we study the existence and uniqueness of the strong solution of following d dimensional stochastic differential equation (SDE) driven by Brownian motion: dX(t)=b(t,X(t))dt+a(t,X(t))dB(t), X(0)= x, where B is a d-dimensional standard Brownian motion; the diffusion coefficient a is a Holder continuous and uniformly non-degenerate matrix-valued function and the drift coefficient b may be discontinuous and unbounded, not necessarily in Sobolev space, extending the previous works to discontinuous and unbounded drift coefficient situation. The idea is to combine the Zvonkin transformation with the Lyapunov function approach. To this end, we need to establish a local version of the connection between the solutions of the SDE up to the exit time of a bounded connected open set D and the associated partial differential equation on this domain. As an interesting byproduct, we establish a localized version of the Krylov estimates (Theorem 4.1) and a localized version of the stability result of the stochastic differential equations of discontinuous coefficients (Theorem 4.5).

math.PR↗
arXiv2024

Carleman estimates for degenerate parabolic equations with single interior point degeneracy and its applications

We study the controllability of a class of $N$-dimensional degenerate parabolic equations with single interior point degeneracy. We employ the Galerkin method to prove the existence of solutions for the equations. The analysis is then divided into two cases based on whether the degenerate point $x=0$ lies within the control region $ω_0$ or not. For each case, we establish specific Carleman estimates. As a result, we achieve null controllability in the first case $0\inω_0$ and unique continuation and approximate controllability in the second case $0\notinω_0$.

math.OC↗
arXiv2024

Asymptotic properties of maximum likelihood estimators for determinantal point processes

We obtain the almost sure strong consistency and the Berry-Esseen type bound for the maximum likelihood estimator Ln of the ensemble L for determinantal point processes (DPPs), strengthening and completing previous work initiated in Brunel, Moitra, Rigollet, and Urschel [BMRU17]. Numerical algorithms of estimating DPPs are developed and simulation studies are performed. Lastly, we give explicit formula and a detailed discussion for the maximum likelihood estimator for blocked determinantal matrix of two by two submatrices and compare it with the frequency method.

math.ST↗
arXiv2024

Null controllability of n-dimensional parabolic equations degenerated on partial boundary

This paper extends the Carleman estimates to high dimensional parabolic equations with highly degenerate symmetric coefficients on a bounded domain of Lipschitz boundary and use these estimates to study the controlla?bility the corresponding equations. Due to the nonsmoothness and degeneracy of boundary, the partial integration by parts in Carleman estimates have no meaning on the degenerate and nonsmooth parts of the boundary. To get around of this difficulty, we construct special weight function, and transform some integral terms in degenerate regions into a non-degenerate ones carefully so that the obtained Carleman estimates can still be used to the controllability problem. Our results includes some well-known works as some special cases as well as some interesting new examples.

math.AP↗