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Chang-Song Deng

Publications and source records attributed to Chang-Song Deng.

15 recordsLinked to original sources
arXiv2024

Well-Posedness for McKean-Vlasov SDEs Driven by Multiplicative Stable Noises

We establish the well-posedness for a class of McKean-Vlasov SDEs driven by symmetric $α$-stable Lévy process ($1/2<α\leq1$), where the drift coefficient is Hölder continuous in space variable, while the noise coefficient is Lipscitz continuous in space variable, and both of them satisfy the Lipschitz condition in distribution variable with respect to Wasserstein distance. If the drift coefficient does not depend on distribution variable, our methodology developed in this paper applies to the case $α\in(0,1]$. The main tool relies on heat kernel estimates for (distribution independent) stable SDEs and Banach's fixed point theorem.

math.PR↗
arXiv2023

Well-Posedness for McKean-Vlasov SDEs with Distribution Dependent Stable Noises

The well-posedness is established for McKean-Vlasov SDEs driven by $α$-stable noises ($1<α<2$). In this model, the drift is Hölder continuous in space variable and Lipschitz continuous in distribution variable with respect to the sum of Wasserstein and weighted variation distances, while the noise coefficient satisfies the Lipschitz condition in distribution variable with respect to the sum of two Wasserstein distances. The main tool relies on Zvonkin's transform, a time-change technique and a two-step fixed point argument.

math.PR↗
arXiv2022

Pathwise Blowup of space-time fractional SPDEs

The finite time blowup in the almost sure sense of a class of space-time fractional stochastic partial differential equations is discussed. Both the cases of white noise and colored noise are considered. The sufficient and necessary condition between the blowup and Osgood condition is obtained when the spatial domain is bounded. And the sufficient condition for the blowup is obtained when the spatial domain is the whole space. The results in this paper could be regarded as extensions to some results in Foondun and Nualart, 2021.

math.PR↗
arXiv2019

Exact Asymptotic Formulas for the Heat Kernels of Space and Time-Fractional Equations

This paper aims to study the asymptotic behaviour of the fundamental solutions (heat kernels) of non-local (partial and pseudo differential) equations with fractional operators in time and space. In particular, we obtain exact asymptotic formulas for the heat kernels of time-changed Brownian motions and Cauchy processes. As an application, we obtain exact asymptotic formulas for the fundamental solutions to the $n$-dimensional fractional heat equations in both time and space \begin{gather*} \frac{\partial^β}{\partial t^β}u(t,x) = -(-Δ_x)^γu(t,x), \quad β,γ\in(0,1). \end{gather*}

math.PR↗
arXiv2019

Semi-implicit Euler-Maruyama method for non-linear time-changed stochastic differential equations

The semi-implicit Euler-Maruyama (EM) method is investigated to approximate a class of time-changed stochastic differential equations, whose drift coefficient can grow super-linearly and diffusion coefficient obeys the global Lipschitz condition. The strong convergence of the semi-implicit EM is proved and the convergence rate is discussed. When the Bernstein function of the inverse subordinator (time-change) is regularly varying at zero, we establish the mean square polynomial stability of the underlying equations. In addition, the numerical method is proved to be able to preserve such an asymptotic property. Numerical simulations are presented to demonstrate the theoretical results.

math.NA↗
arXiv2019

Subgeometric Rates of Convergence for Discrete Time Markov Chains under Discrete Time Subordination

In this note, we are concerned with the subgeometric rate of convergence of a Markov chain with discrete time parameter to its invariant measure in the $f$-norm. We clarify how three typical subgeometric rates of convergence are inherited under a discrete time version of Bochner's subordination. The crucial point is to establish the corresponding moment estimates for discrete time subordinators under some reasonable conditions on the underlying Bernstein function.

math.PR↗
arXiv2017

Harnack Inequalities for SDEs Driven by Time-Changed Fractional Brownian Motions

We establish Harnack inequalities for stochastic differential equations (SDEs) driven by a time-changed fractional Brownian motion with Hurst parameter $H\in(0,1/2)$. The Harnack inequality is dimension-free if the SDE has a drift which satisfies a one-sided Lipschitz condition, otherwise we still get Harnack-type estimates, but the constants will, in general, depend on the space dimension. Our proof is based on a coupling argument and a regularization argument for the time-change.

math.PR↗
arXiv2017

Subgeometric rates of convergence for Markov processes under subordination

We are interested in the rate of convergence of a subordinate Markov process to its invariant measure. Given a subordinator and the corresponding Bernstein function (Laplace exponent) we characterize the convergence rate of the subordinate Markov process; the key ingredients are the rate of convergence of the original process and the (inverse of the) Bernstein function. At a technical level, the crucial point is to bound three types of moments (sub-exponential, algebraic and logarithmic) for subordinators as time $t$ tends to infinity. At the end we discuss some concrete models and we show that subordination can dramatically change the speed of convergence to equilibrium.

math.PR↗
arXiv2016

Complete Bernstein functions and subordinators with nested ranges. A note on a paper by P. Marchal

Let $α:[0,1]\to [0,1]$ be a measurable function. It was proved by P. Marchal \cite{Mar15} that the function $$ ϕ^{(α)}(λ):=\exp\left[ \int_0^1\frac{λ-1}{1+(λ-1)x}\,α(x)\,d x \right],\quad λ>0 $$ is a special Bernstein function. Marchal used this to construct, on a single probability space, a family of regenerative sets $\mathcal R^{(α)}$ such that $\mathcal{R}^{(α)} \stackrel{\text{law}}{=} \overline{\{S^{(α)}_t:t\geq 0\}}$ ($S^{(α)}$ is the subordinator with Laplace exponent $ϕ^{(α)}$) and $\mathcal R^{(α)}\subset \mathcal R^{(β)}$ whenever $α\leqβ$. We give two simple proofs showing that $ϕ^{(α)}$ is a complete Bernstein function and extend Marchal's construction to all complete Bernstein functions.

math.PR↗