Terence Tao, Van Vu
Let $M$ be an arbitrary $n$ by $n$ matrix. We study the condition number a random perturbation $M+N_n$ of $M$, where $N_n$ is a random matrix. It is shown that, under very general conditions on $M$ and $M_n$, the condition number of $M+N_n$ is polynomial in $n$ with very high probability. The main novelty here is that we allow $N_n$ to have discrete distribution.
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Optimal matchings of randomly perturbed lattices
本文针对随机扰动格点过程，构造了与格之间的平移不变完美匹配，证明匹配距离尾部与空穴概率一致，达到自然下界。
We derive a sufficient condition for stability in probability of an equilibrium of a randomly perturbed map in ${\mathbb R}^d$. This condition can be used to stabilize weakly unstable equilibria by random forcing. Analytical results on stabilization are illustrated with numerical examples of randomly perturbed linear and nonlinear maps in one- and two-dimensional spaces.
Fluctuation analysis for a randomly perturbed dynamical system with state-dependent impulse effects
本文研究了随机扰动下平面冲击动力系统的极限定理，通过极坐标简化示例，推导了径向分量的小噪声展开及Skorohod空间中的误差估计。
We consider the average number of limit cycles that bifurcate from a randomly perturbed linear center where the perturbation consists of random (bivariate) polynomials with independent coefficients. This problem reduces, by way of classical perturbation theory of the Poincaré first return map, to a problem on the real zeros of a random \emph{univariate} polynomial $\displaystyle f_n(x) = \sum_{m=0}^n c_m ξ_m x^m$ with independent coefficients $ξ_m$ having mean zero, variance 1 and $c_m \sim m^{-1/2}$. This polynomial belongs to the class of {\it generalized Kac polynomials} at the critical reg…
Iteration of randomly chosen quadratic maps defines a Markov process: X_{n+1}=ε_{n+1}X_n(1-X_n), where ε_n are i.i.d. with values in the parameter space [0,4] of quadratic maps F_θ(x)=θx(1-x). Its study is of significance as an important Markov model, with applications to problems of optimization under uncertainty arising in economics. In this article a broad criterion is established for positive Harris recurrence of X_n.
We consider incompressible generalized Newtonian fluids in two space dimensions perturbed by an additive Gaussian noise. The velocity field of such a fluid obeys a stochastic partial differential equation with fully nonlinear drift due to the dependence of viscosity on the shear rate. In particular, we assume that the extra stress tensor is of power law type, i.\,e. a polynomial of degree $p-1$, $p \in (1,2)$, i.\,e. the shear thinning case. We prove that the associated Kolmogorov operator $K$ admits at least one infinitesimally invariant measure $μ$ satisfying certain exponential moment estim…
Convergence properties of random ergodic averages have been extensively studied in the literature. In these notes, we exploit a uniform estimate by Cohen \& Cuny who showed convergence of a series along randomly perturbed times for functions in $L^2$ with $\int \max(1,\log (1+|t|)) dμ_f<\infty$. We prove universal pointwise convergence of a class of random averages along randomly perturbed times for $L^2$ functions with $\int \max(1,\log\log(1+|t|)) dμ_f<\infty$. For averages with additional smoothing properties, we obtain a universal variational inequality as well as universal pointwise conve…
Limiting Behavior of Randomly Perturbed Diffusions with Invariant Repelling Surfaces
研究随机扰动扩散过程在存在不变排斥面时的极限行为，揭示了不同时间尺度下亚稳分布与遍历不变测度的关系。
The interior of randomly perturbed self-similar sets on the line
本文针对一类随机自相似集否定了“存在正测度空内部直线自相似集”的问题，并证明了关于这类集合代数差性质的Palis-Takens猜想。
We consider a multidimensional time-homogeneous dynamical system and add a randomly perturbed time-dependent deterministic signal to some of its components, giving rise to a high-dimensional system of stochastic differential equations, which is driven by possibly very low-dimensional noise. Equations of this type commonly occur in biology when modelling neurons or in statistical mechanics for certain Hamiltonian systems. We provide verifiable conditions on the original deterministic dynamical system under which the solution to the respective stochastic system features a point in the interior o…
In this paper, we study the effect of small Brownian noise on a switching dynamical system which models a first-order DC/DC buck converter. The state vector of this system comprises a continuous component whose dynamics switch, based on the ON/OFF configuration of the circuit, between two ordinary differential equations (ODE), and a discrete component which keeps track of the ON/OFF configurations. Assuming that the parameters and initial conditions of the unperturbed system have been tuned to yield a stable periodic orbit, we study the stochastic dynamics of this system when the forcing input…
On the Color Discrepancy of Spanning Trees in Random and Randomly Perturbed Graphs
本文研究随机图和随机扰动稠密图中生成树的颜色偏差，证明在连通阈值以上的随机图及添加少量随机边的稠密图中，任意2-边着色下都存在具有较大颜色偏差的生成树。
The asymptotic behavior of the integrated density of states for a randomly perturbed lattice at the infimum of the spectrum is investigated. The leading term is determined when the decay of the single site potential is slow. The leading term depends only on the classical effect from the scalar potential. To the contrary, the quantum effect appears when the decay of the single site potential is fast. The corresponding leading term is estimated and the leading order is determined. In the multidimensional cases, the leading order varies in different ways from the known results in the Poisson case…
A system of autonomous differential equations with a stable limit cycle and perturbed by small white noise is analyzed in this work. In the vicinity of the limit cycle of the unperturbed deterministic system, we define, construct, and analyze the Poincare map of the randomly perturbed periodic motion. We show that the time of the first exit from a small neighborhood of the fixed point of the map, which corresponds to the unperturbed periodic orbit, is well approximated by the geometric distribution. The parameter of the geometric distribution tends zero together with the noise intensity. There…
Efficient uncertainty quantification for mechanical properties of randomly perturbed elastic rods
针对随机几何扰动下弹性杆有效模量的不确定性量化问题，本文通过细长杆Γ-极限推导一维替代模型，建立随机均匀化结果，并结合有限元与蒙特卡罗模拟验证其有效性。
This paper is concerned with backward problem for nonlinear space fractional diffusion with additive noise on the right-hand side and the final value. To regularize the instable solution, we develop some new regularized method for solving the problem. In the case of constant coefficients, we use the truncation methods. In the case of perturbed time dependent coefficients, we apply a new quasi-reversibility method. We also show the convergence rate between the regularized solution and the sought solution under some a priori assumption on the sought solution.
The zero-range process is a stochastic interacting particle system that exhibits a condensation transition under certain conditions on the dynamics. It has recently been found that a small perturbation of a generic class of jump rates leads to a drastic change of the phase diagram and prevents condensation in an extended parameter range. We complement this study with rigorous results on a finite critical density and quenched free energy in the thermodynamic limit, as well as quantitative heuristic results for small and large noise which are supported by detailed simulation data. While our new …
We analyze quasi-stationary distributions $\{μ^{\varepsilon}\}_{\varepsilon>0}$ of a family of Markov chains $\{X^{\varepsilon}\}_{\varepsilon>0}$ that are random perturbations of a bounded, continuous map $F:M\to M$, where $M$ is a closed subset of $\mathbb{R}^k$. Consistent with many models in biology, these Markov chains have a closed absorbing set $M_0\subset M$ such that $F(M_0)=M_0$ and $F(M\setminus M_0)=M\setminus M_0$. Under some large deviations assumptions on the random perturbations, we show that, if there exists a positive attractor for $F$ (i.e., an attractor for $F$ in $M\setmin…
Study of random networks generally requires the nodes to be independently and uniformly distributed such as a Poisson point process. In this work, we venture beyond this standard paradigm and investigate a stochastic forest obtained from a drainage network model constructed on a randomly perturbed subset of $\mathbb{Z}^2$, where both horizontal and vertical perturbations are given by exponentially decaying unbounded discrete random variables and vertical perturbations are allowed in the upward direction only. We show that the resultant stochastic network is a single tree a.s. We further establ…
We consider the Gibbs-measures of continuous-valued height configurations on the $d$-dimensional integer lattice in the presence a weakly disordered potential. The potential is composed of Gaussians having random location and random depth; it becomes periodic under shift of the interface perpendicular to the base-plane for zero disorder. We prove that there exist localized interfaces with probability one in dimensions $d\geq 3+1$, in a `low-temperature' regime. The proof extends the method of continuous-to-discrete single- site coarse graining that was previously applied by the author for a do…
We study irreducible time-homogenous Markov chains with finite state space in discrete time. We obtain results on the sensitivity of the stationary distribution and other statistical quantities with respect to perturbations of the transition matrix. We define a new closeness relation between transition matrices, and use graph-theoretic techniques, in contrast with the matrix analysis techniques previously used.
Surface waves in randomly perturbed discrete models
本文基于离散 Gurtin-Murdoch 模型分析随机局域质量扰动下的表面波传播，推导反射、透射与辐射损失统计，并揭示弱、强散射区的不同标度规律。
A probabilistic Weyl-law for perturbed Berezin-Toeplitz operators
本文证明随机扰动Berezin-Toeplitz算子谱的概率Weyl律，推广Vogel 2020年结果，采用其策略及作者发展的奇异符号演算。
The parabolic Anderson problem with a random potential obtained by attaching a long tailed potential around a randomly perturbed lattice is studied. The moment asymptotics of the total mass of the solution is derived.The results show that the total mass of the solution concentrates on a small set in the space of configuration.
We consider binary infinite order stochastic chains perturbed by a random noise. This means that at each time step, the value assumed by the chain can be randomly and independently flipped with a small fixed probability. We show that the transition probabilities of the perturbed chain are uniformly close to the corresponding transition probabilities of the original chain. As a consequence, in the case of stochastic chains with unbounded but otherwise finite variable length memory, we show that it is possible to recover the context tree of the original chain, using a suitable version of the alg…
We study perturbations of topological pressures, Gibbs measures and measure-theoretic entropies of these measures concerning perturbed potentials defined on topologically transitive subshift of finite type. The subshift with respect to non-perturbed system is assumed to be no topologically transitive. Therefore, the subshift of the perturbed systems and the subshift of the unperturbed system are different. We reduce this situation to a perturbation problem of certain irreducible nonnegative matrices generated by Ruelle transfer operators. Consequently, under suitable conditions of potentials, …
In this paper, we find a regularized approximate solution for an inverse problem for the Burgers' equation. The solution of the inverse problem for the Burgers' equation is ill-posed, i.e., the solution does not depend continuously on the data. The approximate solution is the solution of a regularized equation with randomly perturbed coefficients and randomly perturbed final value and source functions. To find the regularized solution, we use the modified quasi-reversibility method associated with the truncated expansion method with nonparametric regression. We also investigate the convergence…
Local large deviations for randomly forced nonlinear wave equations with localized damping
针对有界噪声扰动的局部阻尼非线性波动方程，本文利用渐近紧性与渐近指数胎紧性建立经验分布的局部型LDP，并提出随机动力系统的新LDP方法。
The perturbed GUE corners ensemble is the joint distribution of eigenvalues of all principal submatrices of a matrix $G+\mathrm{diag}(\mathbf{a})$, where $G$ is the random matrix from the Gaussian Unitary Ensemble (GUE), and $\mathrm{diag}(\mathbf{a})$ is a fixed diagonal matrix. We introduce Markov transitions based on exponential jumps of eigenvalues, and show that their successive application is equivalent in distribution to a deterministic shift of the matrix. This result also leads to a new distributional symmetry for a family of reflected Brownian motions with drifts coming from an arith…
We study the spectra of general $N\times N$ Toeplitz matrices given by symbols in the Wiener Algebra perturbed by small complex Gaussian random matrices, in the regime $N\gg 1$. We prove an asymptotic formula for the number of eigenvalues of the perturbed matrix in smooth domains. We show that these eigenvalues follow a Weyl law with probability sub-exponentially close to $1$, as $N\gg1$, in particular that most eigenvalues of the perturbed Toeplitz matrix are close to the curve in the complex plane given by the symbol of the unperturbed Toeplitz matrix.
We study a class of rotation invariant determinantal ensembles in the complex plane; examples include the eigenvalues of Gaussian random matrices and the roots of certain families of random polynomials. The main result is a criteria for a central limit theorem to hold for angular statistics of the points. The proof exploits an exact formula relating the generating function of such statistics to the determinant of a perturbed Toeplitz matrix.
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