arXiv subjects
Publications and source records attributed to Lihu Xu.
A Freidlin-Wentzell type large deviation principle is established for stochastic partial differential equations with slow and fast time-scales, where the slow component is a one-dimensional stochastic Burgers equation with small noise and the fast component is a stochastic reaction-diffusion equation. Our approach is via the weak convergence criterion developed in [3].
By the Lindeberg principle, we develop in this paper an approximation to one dimensional (possibly) asymmetric $α$-stable distributions with $α\in (0,2)$ in the smooth Wasserstein distance. It is the first time that the general stable central limit theorem is proved by the Lindeberg principle, and that this theorem with $α\in (0,1]$ is proved by a new method other than Fourier analysis. Our main tools are a Taylor-like expansion and a Kolmogorov forward equation.
We develop Stein's method for $α$-stable approximation with $α\in(0,1]$, continuing the recent line of research by Xu \cite{lihu} and Chen, Nourdin and Xu \cite{C-N-X} in the case $α\in(1,2).$ The main results include an intrinsic upper bound for the error of the approximation in a variant of Wasserstein distance that involves the characterizing differential operators for stable distributions, and an application to the generalized central limit theorem. Due to the lack of first moment for the approximating sequence in the latter result, we appeal to an additional truncation procedure and investigate fine regularity properties of the solution to Stein's equation.
Let $n \in \mathbb N$, let $ζ_{n,1},...,ζ_{n,n}$ be a sequence of independent random variables with $\mathbb E ζ_{n,i}=0$ and $\mathbb E |ζ_{n,i}|<\infty$ for each $i$, and let $μ$ be an $α$-stable distribution having characteristic function $e^{-|λ|^α}$ with $α\in (1,2)$. Denote $S_{n}=ζ_{n,1}+...+ζ_{n,n}$ and its distribution by $\mathcal L(S_n)$, we bound the Wasserstein distance of $\mathcal L(S_{n})$ and $μ$ essentially by an $L^{1}$ discrepancy between two kernels, this bound can be interpreted as a generalization of the Stein discrepancy (in $L^{2}$ sense) introduced by Ledoux, Nourdin and Peccati. More precisely, we prove the following inequality: \begin{equation} \begin{split} d_W\left(\mathcal L (S_n), μ\right) \ \le C \left[\sum_{i=1}^n\int_{-N}^N \left|\frac{\mathcal K_α(t,N)}n -\frac{ K_i(t,N)}α\right| d t \ +\ \mathcal R_{N,n}\right], \end{split} \end{equation} where $d_{W}$ is the Wasserstein distance of probability measures, $\mathcal K_α(t,N)$ is the kernel of a decomposition of the fractional Laplacian $Δ^{\frac \alpha2}$, $ K_i(t,N)$ is a kernel introduced by Chen, Goldstein and Shao with a truncation which can be interpreted as an $L^1$ Stein kernel, and $\mathcal R_{N,n}$ is a small remainder. The integral term $$\sum_{i=1}^n\int_{-N}^N \left|\frac{\mathcal K_α(t,N)}n -\frac{ K_i(t,N)}α\right| d t$$ can be interpreted as an $L^{1}$ Stein discrepancy. As an application, we prove a general theorem of stable law convergence rate when $ζ_{n,i}$ are i.i.d. and the distribution falls in the normal domain of attraction of $μ$. We also study four examples with comparing our convergence rates and those known for these examples, among which the distribution in the second example is not in the normal domain of attraction of $μ$.
This paper is concerned with the Stein's method associated with a (possibly) asymmetric $α$-stable distribution $Z$, in dimension one. More precisely, its goal is twofold. In the first part, we exhibit a genuine bound for the Wasserstein distance between $Z$ and any integrable random variable $X$, in terms of an operator that reduces to the classical fractional Laplacian in the symmetric case. Then, in the second part we apply the aforementioned bound to compute error rates in the stable central limit theorem, when the entries are in the domain $\mathcal{D}_α$ of normal attraction of a stable law of exponent $α$. To conclude, we study the specific case where the entries are Pareto like multiplied by a slowly varying function, which provides an example of random variables that do not belong to $\mathcal{D}_α$, but for which our approach continues to apply.
Stein's method has been widely used for probability approximations. However, in the multi-dimensional setting, most of the results are for multivariate normal approximation or for test functions with bounded second- or higher-order derivatives. For a class of multivariate limiting distributions, we use Bismut's formula in Malliavin calculus to control the derivatives of the Stein equation solutions by the first derivative of the test function. Combined with Stein's exchangeable pair approach, we obtain a general theorem for multivariate approximations with near optimal error bounds on the Wasserstein distance.We apply the theorem to the unadjusted Langevin algorithm.
It is well known that $\int_{0}^{1} t^{-θ} d t<\infty$ for $θ\in (0,1)$ and $\int_{0}^{1} t^{-θ} d t=\infty$ for $θ\in [1,\infty)$. Since $t$ can be taken as an $α$-stable subordinator with $α=1$, it is natural to ask whether $\int_{0}^{1} t^{-θ} d S_{t}$ has a similar property when $S_{t}$ is an $α$-stable subordinator with $α\in (0,1)$. We show that $θ=\frac 1α$ is the border line such that $\int_{0}^{1} t^{-θ} d S_{t}$ is finite a.s. for $θ\in (0, \frac 1α)$ and blows up a.s. for $θ\in [\frac1α,\infty)$. When $α=1$, our result recovers that of $\int_{0}^{1} t^{-θ} d t$. Moreover, we give a $p$-th moment estimate for the integral when $θ\in (0,\frac 1α)$.
In this paper we prove uniform Hausdorff and packing dimension results for the images of a large family of Markov processes. The main tools are the two covering principles of Xiao (second author). As applications, uniform Hausdorff and packing dimension results for certain classes of Lévy processes, stable jump diffusion and non-symmetric stable-like processes are obtained.
Using heat kernel estimates, we prove the pathwise uniqueness for strong solutions of irregular stochastic differential equation driven by a family of Markov process, whose generator is a non-local and non-symmetric Lévy type operator. Due to the extra term $1_{[0,σ(X_{s-},z)]}(r)$ in multiplicative noise, we need to derive some new regularity results for the generator and use a trick of mixing $L_1$ and $L_2$-estimates by Kurtz and Protter \cite{Ku-Po}.
We proposed a new penalized method in this paper to solve sparse Poisson Regression problems. Being different from $\ell_1$ penalized log-likelihood estimation, our new method can be viewed as penalized weighted score function method. We show that under mild conditions, our estimator is $\ell_1$ consistent and the tuning parameter can be pre-specified, which shares the same good property of the square-root Lasso.
We study the ergodicity of stochastic reaction-diffusion equation driven by subordinate Brownian motions. After establishing the strong Feller property and irreducibility of the system, we prove the tightness of the solution's law. These properties imply that this stochastic system admits a unique invariant measure according to Doob's and Krylov-Bogolyubov's theories. Furthermore, we establish a large deviation principle for the occupation measure of this system by a hyper-exponential recurrence criterion. It is well known that S(P)DEs driven by $α$-stable type noises do not satisfy Freidlin-Wentzell type large deviation, our result gives an example that strong dissipation overcomes heavy tailed noises to produce a Donsker-Varadhan type large deviation as time tends to infinity.
Using the hyper-exponential recurrence criterion, a large deviation principle for the occupation measure is derived for a class of non-linear monotone stochastic partial differential equations. The main results are applied to many concrete SPDEs such as stochastic $p$-Laplace equation, stochastic porous medium equation, stochastic fast-diffusion equation, and even stochastic real Ginzburg-Landau equation driven by $α$-stable noises.
Under some regularity conditions on $b$, $σ$ and $α$, we prove that the following perturbed stochastic differential equation \begin{equation} X_t=x+\int_0^t b(X_s)ds+\int_0^t σ(X_s) dB_s+α\sup_{0 \le s \le t} X_s, \ \ \ α<1 \end{equation} admits smooth densities for all $0 \le t \le t_0$, where $t_0>0$ is some finite number.
We establish a large deviation principle for the occupation measure of the stochastic real Ginzburg-Landau equation driven by $α$-stable noises. The proof is based on a hyper-exponential recurrence criterion. Our result indicates a phenomenon that strong dissipation beats heavy tailed noises to produce a large deviation, it seems to us that this phenomenon has not been reported in the known literatures.
We establish the irreducibility of stochastic real Ginzburg-Landau equation with $α$-stable noises by a maximal inequality and solving a control problem. As applications, we prove that the system converges to its equilibrium measure with exponential rate under a topology stronger than total variation and obeys the moderate deviation principle by constructing some Lyapunov test functions.
By using the dimension-free Harnack inequality and the integration by parts formula for the associated diffusion semigroup, we prove the central limit theorem, the moderate deviation principle, and the logarithmic iteration law for the sample entropy production rate of stochastic differential equations with Lipschitz continuous and dissipative drifts.
Fluctuation theorem is one of the major achievements in the field of nonequilibrium statistical mechanics during the past two decades. Steady-state fluctuation theorem of sample entropy production rate in terms of large deviation principle for diffusion processes have not been rigorously proved yet due to technical difficulties. Here we give a proof for the steady-state fluctuation theorem of a diffusion process in magnetic fields, with explicit expressions of the free energy function and rate function. The proof is based on the Karhunen-Loéve expansion of complex-valued Ornstein-Uhlenbeck process.
By a coupling method, we prove that a family of stochastic partial differential equations (SPDEs) driven by highly degenerate pure jump Lévy noises are exponential mixing. These pure jump Lévy noises include $α$-stable process with $α\in (0,2)$