We study the parabolic Anderson problem, that is, the heat equation $\partial_tu=Δu+ξu$ on $(0,\infty)\times{\mathbb{Z}}^d$ with independent identically distributed random potential $\{ξ(z):z\in{\mathbb{Z}}^d\}$ and localized initial condition $u(0,x)={\mathbf{1}}_0(x)$. Our interest is in the long-term behavior of the random total mass $U(t)=\sum_zu(t,z)$ of the unique nonnegative solution in the case that the distribution of $ξ(0)$ is heavy tailed. For this, we study two paradigm cases of distributions with infinite moment generating functions: the case of polynomial or Pareto tails, and the case of stretched exponential or Weibull tails. In both cases we find asymptotic expansions for the logarithm of the total mass up to the first random term, which we describe in terms of weak limit theorems. In the case of polynomial tails, already the leading term in the expansion is random. For stretched exponential tails, we observe random fluctuations in the almost sure asymptotics of the second term of the expansion, but in the weak sense the fourth term is the first random term of the expansion. The main tool in our proofs is extreme value theory.
We prove that the almost sure Lyapunov exponent λ(κ) of the continuous space Parabolic Anderson Model is bounded above by $c_u κ^{1/3}$ as $κ\downarrow0$ under mild regularity conditions. This bound of the same order of the previously proven lower bound, $λ(κ) \ge c_l κ^{1/3}$.
We consider a discrete-time version of the parabolic Anderson model. This may be described as a model for a directed (1+d)-dimensional polymer interacting with a random potential, which is constant in the deterministic direction and i.i.d. in the d orthogonal directions. The potential at each site is a positive random variable with a polynomial tail at infinity. We show that, as the size of the system diverges, the polymer extremity is localized almost surely at one single point which grows ballistically. We give an explicit characterization of the localization point and of the typical paths o…
We find the asymptotics for the almost sure Lyapunov exponent for the solution of the parabolic Anderson problem as the molecular diffusivity tends to zero.
Consider a heavy-tailed branching process (denoted by $Z_{n}$) in random environments, under the condition which infers that $\mathbb{E}\log m(ξ_{0})=\infty$. We show that (1) there exists no proper $c_{n}$ such that $\{Z_{n}/c_{n}\}$ has a proper, non-degenerate limit, (2) normalized by a sequence of functions, a proper limit can be obtained, i.e., $y_{n}\left(\barξ,Z_{n}(\barξ)\right)$ converges almost surely to a random variable $Y(\barξ)$, where $Y\in(0,1)~η$-a.s., (3) finally, we give a necessary and sufficient conditions for the almost sure convergence of $\left\{\frac{U(\barξ,Z_{n}(\bar…
We consider the Anderson polymer partition function $$ u(t):=\mathbb{E}^X\Bigl[e^{\int_0^t \mathrm{d}B^{X(s)}_s}\Bigr]\,, $$ where $\{B^{x}_t\,;\, t\geq0\}_{x\in\mathbb{Z}^d}$ is a family of independent fractional Brownian motions all with Hurst parameter $H\in(0,1)$, and $\{X(t)\}_{t\in \mathbb{R}^{\geq 0}}$ is a continuous-time simple symmetric random walk on $\mathbb{Z}^d$ with jump rate $κ$ and started from the origin. $\mathbb{E}^X$ is the expectation with respect to this random walk. We prove that when $H\leq 1/2$, the function $u(t)$ almost surely grows asymptotically like $e^{l t}$, wh…
In this paper we consider the Anderson Hamiltonian with white noise potential on the box $[0,L]^2$ with Dirichlet boundary conditions. We show that all the eigenvalues divided by $\log L$ converge as $L\rightarrow \infty$ almost surely to the same deterministic constant, which is given by a variational formula.
Marcinkiewicz strong law of large numbers, ${n^{-\frac1p}}\sum_{k=1}^{n} (d_{k}- d)\rightarrow 0\ $ almost surely with $p\in(1,2)$, are developed for products $d_k=\prod_{r=1}^s x_k^{(r)}$, where the $x_k^{(r)} = \sum_{l=-\infty}^{\infty}c_{k-l}^{(r)}ξ_l^{(r)}$ are two-sided linear processes with coefficients $\{c_l^{(r)}\}_{l\in \mathbb{Z}}$ and i.i.d. zero-mean innovations $\{ξ_l^{(r)}\}_{l\in \mathbb{Z}}$. The decay of the coefficients $c_l^{(r)}$ as $|l|\to\infty$, can be slow enough for $\{x_k^{(r)}\}$ to have long memory while $\{d_k\}$ can have heavy tails. The long-range dependence and…
We consider the Feynman-Kac functional associated with a Brownian motion in a random potential. The potential is defined by attaching a heavy tailed positive potential around the Poisson point process. This model was first considered by Pastur (1977) and the first order term of the moment asymptotics was determined. In this paper, both moment and almost sure asymptotics are determined up to the second order. As an application, we also derive the second order asymptotics of the integrated density of states of the corresponding random Schrödinger operator.
The aim of this paper is to establish the almost sure asymptotic behavior as the space variable becomes large, for the solution to the one spatial dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance structure of a fractional Brownian motion with Hurst parameter greater than 1/4 and less than 1/2 in the space variable.
The parabolic Anderson problem is the Cauchy problem for the heat equation with random potential and localized initial condition. In this paper we consider potentials which are constant in time and independent exponentially distributed in space. We study the growth rate of the total mass of the solution in terms of weak and almost sure limit theorems, and the spatial spread of the mass in terms of a scaling limit theorem. The latter result shows that in this case, just like in the case of heavy tailed potentials, the mass gets trapped in a single relevant island with high probability.
The parabolic Anderson model is the heat equation with some extra spatial randomness. In this paper we consider the parabolic Anderson model with i.i.d. Pareto potential on a critical Galton-Watson tree conditioned to survive. We prove that the solution at time $t$ is concentrated at a single site with high probability and at two sites almost surely as $t \to \infty$. Moreover, we identify asymptotics for the localisation sites and the total mass, and show that the solution $u(t,v)$ at a vertex $v$ can be well-approximated by a certain functional of $v$. The main difference with earlier result…
The parabolic Anderson model on $\mathbb{Z}^d$ with i.i.d. potential is known to completely localise if the distribution of the potential is sufficiently heavy-tailed at infinity. In this paper we investigate a modification of the model in which the potential is partially duplicated in a symmetric way across a plane through the origin. In the case of potential distribution with polynomial tail decay, we exhibit a surprising phase transition in the model as the decay exponent varies. For large values of the exponent the model completely localises as in the i.i.d. case. By contrast, for small va…
The parabolic Anderson model is the Cauchy problem for the heat equation with a random potential. We consider this model in a setting which is continuous in time and discrete in space, and focus on time-constant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the long-term temporal dynamics of this system. Our main result is that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time, a phenomenon known as ageing. We describe this phenomenon in the weak sense, by looking at the a…
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.
Let $B_s$ be a $d$-dimensional Brownian motion and $ω(dx)$ be an independent Poisson field on $\mathbb{R}^d$. The almost sure asymptotics for the logarithmic moment generating function [\log\math bb{E}_0\exp\biggl{\pmθ\int_0^t\bar{V}(B_s) ds\biggr}\qquad (t\to\infty)] are investigated in connection with the renormalized Poisson potential of the form [\bar{V}(x)=\int_{\mathbb{R}^d}{\frac{1}{|y-x|^p}}[ω(dy)-dy],\qquad x\in\mathbb{R}^d.] The investigation is motivated by some practical problems arising from the models of Brownian motion in random media and from the parabolic Anderson models.
We study asymptotic properties of spatially non-homogeneous random walks with non-integrable increments, including transience, almost-sure bounds, and existence and non-existence of moments for first-passage and last-exit times. In our proofs we also make use of estimates for hitting probabilities and large deviations bounds. Our results are more general than existing results in the literature, which consider only the case of sums of independent (typically, identically distributed) random variables. We do not assume the Markov property. Existing results that we generalize include a circle of i…
The parabolic Anderson problem is the Cauchy problem for the heat equation $\partial_tu(t,z)=Δu(t,z)+ξ(z)u(t,z)$ on $(0,\infty)\times {\mathbb{Z}}^d$ with random potential $(ξ(z):z\in{\mathbb{Z}}^d)$. We consider independent and identically distributed potentials, such that the distribution function of $ξ(z)$ converges polynomially at infinity. If $u$ is initially localized in the origin, that is, if $u(0,{z})={\mathbh1}_0({z})$, we show that, as time goes to infinity, the solution is completely localized in two points almost surely and in one point with high probability. We also identify the …
We consider the random Schrödinger operator on $\mathbb{R}$ obtained by perturbing the Laplacian with a white noise. We prove that Anderson localization holds for this operator: almost surely the spectral measure is pure point and the eigenfunctions are exponentially localized. We give two separate proofs of this result. We also present a detailed construction of the operator and relate it to the parabolic Anderson model. Finally, we discuss the case where the noise is smoothed out.
Let $X_N$ be an $N\ts N$ random symmetric matrix with independent equidistributed entries. If the law $P$ of the entries has a finite second moment, it was shown by Wigner \cite{wigner} that the empirical distribution of the eigenvalues of $X_N$, once renormalized by $\sqrt{N}$, converges almost surely and in expectation to the so-called semicircular distribution as $N$ goes to infinity. In this paper we study the same question when $P$ is in the domain of attraction of an $α$-stable law. We prove that if we renormalize the eigenvalues by a constant $a_N$ of order $N^{\frac{1}α}$, the correspo…
We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure $μ$ of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N by N symmetric matrix $Y_N^σ$ whose (i,j) entry is $σ(i/N,j/N)X_{ij}$ where $(X_{ij}, 0<i<j+1<\infty)$ is an infinite array of i.i.d real variables with common distribution in the domain of attraction of an $α$-stable law, $0<α<2$, and $σ$ is a deterministic function. For a random diagonal $D_N$ independent of $Y_N^σ$ and with appropriate rescaling $a_N$, we prove that the distribution $μ$ of $a_…
We study the multi-particle Anderson model in the continuum and show that under some mild assumptions on the random external potential and the inter-particle interaction, for any finite number of particles, the multi-particle lower edges of the spectrum are almost surely constant in absence of ergodicity. We stress that this result is not quite obvious and has to be handled carefully. In addition, we prove the spectral exponential and the strong dynamical localization of the continuous multi-particle Anderson model at low energy. The proof based on the multi-particle multi-scale analysis bound…
We consider the parabolic Anderson problem $\partial_t u=κΔu+ξu$ on $(0,\infty)\times \Z^d$ with random i.i.d. potential $ξ=(ξ(z))_{z\in\Z^d}$ and the initial condition $u(0,\cdot)\equiv1$. Our main assumption is that $\esssupξ(0)=0$. Depending on the thickness of the distribution $\prob(ξ(0)\in\cdot)$ close to its essential supremum, we identify both the asymptotics of the moments of $u(t,0)$ and the almost-sure asymptotics of $u(t,0)$ as $t\to\infty$ in terms of variational problems. As a by-product, we establish Lifshitz tails for the random Schrödinger operator $-κΔ-ξ$ at the bottom of its…
We consider the large-time behavior of the solution $u\colon [0,\infty)\times\Z\to[0,\infty)$ to the parabolic Anderson problem $\partial_t u=κΔu+ξu$ with initial data $u(0,\cdot)=1$ and non-positive finite i.i.d. potentials $(ξ(z))_{z\in\Z}$. Unlike in dimensions $d\ge2$, the almost-sure decay rate of $u(t,0)$ as $t\to\infty$ is not determined solely by the upper tails of $ξ(0)$; too heavy lower tails of $ξ(0)$ accelerate the decay. The interpretation is that sites $x$ with large negative $ξ(x)$ hamper the mass flow and hence screen off the influence of more favorable regions of the potential…
For the multi-particle Anderson model with correlated random potential in the continuum, we show under fairly general assumptions on the inter-particle interaction and the random external potential, the Anderson localization which consists of both the spectral, exponential localization and the strong dynamical localization. The localization results are proven near the lower spectral edge of the almost sure spectrum and the proofs require the uniform log-Hölder continuity assumption of the probability distribution functions of the random field in addition of the Rosenblatt's strongly mixing con…
On the phase transition for the number of collisions on comb graphs
National University of Singapore（新加坡国立大学）；The Hong Kong University（香港大学）；The Hong Kong University of Science and Technology（香港科技大学）；Technische Universität München（慕尼黑工业大学）
We consider random products of $SL(2, \mathbb{R})$ matrices that depend on a parameter in a non-uniformly hyperbolic regime. We show that if the dependence on the parameter is monotone then almost surely the random product has upper (limsup) Lyapunov exponent that is equal to the value prescribed by the Furstenberg Theorem (and hence positive) for all parameters, but the lower (liminf) Lyapunov exponent is equal to zero for a dense $G_δ$ set of parameters of zero Hausdorff dimension. As a byproduct of our methods, we provide a purely geometrical proof of Spectral Anderson Localization for disc…
Let $X$ and $Y$ be two independent random walks on $\Z^2$ with zero mean and finite variances, and let $L_t(X,Y)$ be the local time of $X-Y$ at the origin at time $t$. We show that almost surely with respect to $Y$, $L_t(X,Y)/\log t$ conditioned on $Y$ converges in distribution to an exponential random variable with the same mean as the distributional limit of $L_t(X,Y)/\log t$ without conditioning. This question arises naturally from the study of the parabolic Anderson model with a single moving catalyst, which is closely related to a pinning model.
The aim of this paper is to study asymptotic geometric properties almost surely or/and in probability of extreme order statistics of an i.i.d. random field (potential) indexed by sites of multidimensional lattice cube, the volume of which unboundedly increases. We discuss the following topics: (I) high level exceedances, in particular, clustering of exceedances; (II) decay rate of spacings in comparison with increasing rate of extreme order statistics; (III) minimum of spacings of successive order statistics; (IV) asymptotic behavior of values neighboring to extremes and so on. The conditions …
Large Deviation Upper Bounds and Improved MSE Rates of Nonlinear SGD: Heavy-tailed Noise and Power of Symmetry
Carnegie Mellon University（卡内基梅隆大学）；Faculty of Technical Sciences, University of Novi Sad（诺维萨德大学技术科学学院）；Faculty of Sciences, University of Novi Sad（诺维萨德大学理学院）
The Tracy-Widom distributions are among the most famous laws in probability theory, partly due to their connection with Wigner matrices. In particular, for $A=\frac{1}{\sqrt{n}}(a_{ij})_{1 \leq i,j \leq n} \in \mathbb{R}^{n \times n}$ symmetric with $(a_{ij})_{1 \leq i \leq j \leq n}$ i.i.d. standard normal, the fluctuations of its largest eigenvalue $λ_1(A)$ are asymptotically described by a real-valued Tracy-Widom distribution $TW_1:$ $n^{2/3}(λ_1(A)-2) \Rightarrow TW_1.$ As it often happens, Gaussianity can be relaxed, and this results holds when $\mathbb{E}[a_{11}]=0, \mathbb{E}[a^2_{11}]=…
