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Shao-Qin Zhang

Publications and source records attributed to Shao-Qin Zhang.

At least 19 recordsLinked to original sources
arXiv2025

A Note on the instability of equilibria for distribution dependent SDEs

Due to the existence of multiple stationary distributions, we study the stability and instability of a stationary distribution for distribution dependent stochastic differential equations. This note is devoted to the instability of a stationary distribution, and links the instability to a spectral property of the generator of the corresponding linearized semigroup to the stochastic equation. Concrete examples, such as the granular media equation with double-wells landscapes, are given to illustrate our main result.

math.PR↗
arXiv2025

Local convergence near equilibria for distribution dependent SDEs

Owing to exhibiting phase transitions, we investigate the local convergence near a stationary distribution for distribution dependent stochastic differential equations. By linearizing the nonlinear Markov semigroup associated with the distribution dependent equation around the stationary distribution, the local exponential convergence of the solution is related to the exponential convergence of a semigroup of linear operators. Our result can be used as a criteria for the locally exponential stability of stationary distributions. Concrete examples, including the granular media equation with double-wells landscapes and quadratic interaction, are given to illustrate our main result.

math.PR↗
arXiv2025

A Local Bifurcation Theorem for McKean-Vlasov Diffusions

We establish an existence result of a solution to a class of probability measure-valued equations, whose solutions can be associated with stationary distributions of many McKean-Vlasov diffusions with gradient-type drifts. Coefficients of the probability measure-valued equation may be discontinuous in the weak topology and the total variation norm. Owing to that the bifurcation point of the probability measure-valued equation is relevant to the phase transition point of the associated McKean-Vlasov diffusion, we establish a local Krasnosel'skii bifurcation theorem. Regularized determinant for the Hilbert-Schmidt operator is used to derive our criteria for the bifurcation point. Concrete examples, including the granular media equation and the Vlasov-Fokker-Planck equation with quadratic interaction, are given to illustrate our results.

math.PR↗
arXiv2024

Distribution dependent SDEs with multiplicative fractional noise

The well-posedness is investigated for distribution dependent stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\in (\ff {\sq 5-1} 2,1)$ and distribution dependent multiplicative noise. To this aim, we introduce a Hölder space of probability measure paths which is a complete metric space under a new metric. Our arguments rely on a mix of contraction mapping principle on the Hölder space and fractional calculus tools. We also establish the large and moderate deviation principles for this type of equations via the weak convergence criteria in the factional Brownian motion setting, which extend previously known results in the additive setting.

math.PR↗
arXiv2024

Exponential convergence in Wasserstein Distance for Diffusion Semigroups with Irregular Drifts

The exponential contraction in $L^1$-Wasserstein distance and exponential convergence in $L^q$-Wasserstein distance ($q\geq 1$) are considered for stochastic differential equations with irregular drift. When the irregular drift drift is locally bounded, the exponential convergence are derived by using the reflection coupling with a new auxiliary function for stochastic differential equations driven by multiplicative noise. Explicit convergence rate is obtained. When the irregular drift is not locally bounded, a new Zvonkin's transformation is given by an ultracontractive reference diffusion, and the exponential convergence is derived by combining the Zvonkin's transformation and related results for stochastic differential equations with locally bounded irregular drift.

math.PR↗
arXiv2022

Exponential convergence in Wasserstein metric for distribution dependent SDEs

The existence and uniqueness of stationary distributions and the exponential convergence in $L^p$-Wasserstein distance are derived for distribution dependent SDEs from associated decoupled equations. To establish the exponential convergence, we introduce a twinned Talagrand inequality of the original SDE and the associated decoupled equation, and explicit convergence rate is obtained. Our results can be applied to SDEs without uniformly dissipative drift and distribution dependent diffusion term, which cover the Curie-Weiss model and the granular media model in double-well landscape with quadratic interaction as examples.

math.PR↗
arXiv2021

Existence and non-uniqueness of stationary distributions for distribution dependent SDEs

The existence of stationary distributions to distribution dependent stochastic differential equations are investigated by using the ergodicity of the associated decoupled equation and the Schauder fixed point theorem. By using Zvonkin's transformation, we also establish the existence result for equations with singular coefficients. Instead of the uniqueness, the non-uniqueness of stationary distributions are considered for equations with regular coefficients. Concrete examples including McKean-Vlasov stochastic equations with the quadratic interaction and the non-quadratic interaction, and equations with a bounded and discontinuous drift are presented to illustrate our non-uniqueness results.

math.PR↗
arXiv2021

A unified approach to gradient type formulas for BSDEs and some applications

In this paper we present a unified approach to establish gradient type formulas and Bismut type formulas for backward stochastic differential equations (BSDEs). This approach relies on a mix of derivative formulas with respect to the conditional probability of forward SDEs and the expression of the solution of BSDEs. Some concrete examples are given to illustrate the results. As applications, we provide representation formulas for the control solutions to McKean-Vlasov BSDEs and derive gradient estimates for related PDEs.

math.PR↗
arXiv2020

Convergence rate of EM algorithm for SDEs under integrability condition

In this paper, by employing Gaussian type estimate of heat kernel, we establish Krylov's estimate and Khasminskill's estimate for EM algorithm. As applications, by taking Zvonkin's transformation into account, we investigate convergence rate of EM algorithm for a class of multidimensional SDEs under integrability conditions, where the drifts need not to be piecewise Lipschitz and are much more singular in some sense.

math.PR↗
arXiv2020

A Zvonkin's transformation for stochastic differential equations with singular drift and related applications

In this paper, by establishing the $L^p$-$L^q$ estimate and Sobolev estimates for parabolic partial differential equations with a singular first order term and a Lipschitz first order term, a new Zvonkin-type transformation is given for stochastic differential equations with singular and Lipschitz drifts. The associated Krylov's estimate is established. As applications, Harnack inequalities are established for stochastic equations with Hölder continuous diffusion coefficient and singular drift term without regularity assumption.

math.PR↗
arXiv2020

TCI for SDEs with irregular drifts

We obtain $T_2(C)$ for stochastic differential equations with Dini continuous drift and $T_1(C)$ stochastic differential equations with singular coefficients.

math.PR↗
arXiv2020

Weak convergence of Euler scheme for SDEs with singular drift

In this paper, we investigate the weak convergence rate of Euler-Maruyama's approximation for stochastic differential equations with irregular drifts. Explicit weak convergence rates are presented if drifts satisfy an integrability condition including discontinuous functions which can be non-piecewise continuous or in fractional Sobolev space.

math.PR↗
arXiv2020

Moment estimates and applications for SDEs driven by fractional Brownian motion with irregular drifts

In this paper, high-order moment, even exponential moment, estimates are established for the Hölder norm of solutions to stochastic differential equations driven by fractional Brownian motion whose drifts are measurable and have linear growth. As applications, we first study the weak uniqueness of solutions to fractional stochastic differential equations. Moreover, combining our estimates and the Fourier transform, we establish the existence of density of solutions to equations with irregular drifts.

math.PR↗
arXiv2020

Exponential contraction in Wasserstein distance on static and evolving manifolds

In this article, exponential contraction in Wasserstein distance for heat semigroups of diffusion processes on Riemannian manifolds is established under curvature conditions where Ricci curvature is not necessarily required to be non-negative. Compared to the results of Wang (2016), we focus on explicit estimates for the exponential contraction rate. Moreover, we show that our results extend to manifolds evolving under a geometric flow. As application, for the time-inhomogeneous semigroups, we obtain a gradient estimate with an exponential contraction rate under weak curvature conditions, as well as uniqueness of the corresponding evolution system of measures.

math.DG↗
arXiv2018

Stochastic differential equations driven by fractional Brownian motion with locally Lipschitiz drift and their Euler approximation

In this paper, we study a class of one-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H>\ff 1 2$. The drift term of the equation is locally Lipschitz and unbounded in the neighborhood of $0$. We show the existence, uniqueness and positivity of the solutions. The estimations of moments, including the negative power moments, are given. Based on these estimations, strong convergence of the positivity preserving drift-implicit Euler-type scheme is proved, and optimal convergence rate is obtained. By using Lamperti transformation, we show that our results can be applied to interest rate models such as mean-reverting stochastic volatility model and strongly nonlinear Aït-Sahalia type model.

math.PR↗
arXiv2018

On invariant probability measures of regime-switching diffusion processes with singular drifts

For regime-switching diffusions processes with singular drifts, we introduce integrability conditions involving a nice reference probability measure and the $Q$-matrix of the jump part to study the existence of the invariant probability measures. Consequently, the generator of the regime-switching diffusions process has an extension in the $L^1$-space w.r.t the invariant probability measure to be a generator of $C_0$-semigroup. Moreover, we prove the uniqueness of the extension. Regularities and the uniqueness of the invariant probability density w.r.t the nice reference probability measure are also considered.

math.PR↗
arXiv2016

Weak Poincaré Inequality for Convolution Probability Measures

By using Lyapunov conditions, weak Poincaré inequalities are established for some probability measures on a manifold $(M,g)$. These results are further applied to the convolution of two probability measures on $\R^d$. Along with explicit results we study concrete examples.

math.PR↗