Abstract:
We present a systematic analytical approach to the trapping of a random walk by a finite density rho of diffusing traps in arbitrary dimension d. We confirm the phenomenologically predicted e^{-c_d rho t^{d/2}} time decay of the survival probability, and compute the dimension dependent constant c_d to leading order within an eps=2-d expansion.

Abstract:
Attributing a positive value \tau_x to each x in Z^d, we investigate a nearest-neighbour random walk which is reversible for the measure with weights (\tau_x), often known as "Bouchaud's trap model". We assume that these weights are independent, identically distributed and non-integrable random variables (with polynomial tail), and that d > 4. We obtain the quenched subdiffusive scaling limit of the model, the limit being the fractional kinetics process. We begin our proof by expressing the random walk as a time change of a random walk among random conductances. We then focus on proving that the time change converges, under the annealed measure, to a stable subordinator. This is achieved using previous results concerning the mixing properties of the environment viewed by the time-changed random walk.

Abstract:
This article is dedicated to localization of the principal eigenvalue (PE) of the Stokes operator acting on solenoidal vector fields that vanish outside a large random domain modeling the pore space in a cubic block of porous material with disordered micro-structure. Its main result is an asymptotically deterministic lower bound for the PE of the sum of a low compressibility approximation to the Stokes operator and a small scaled random potential term, which is applied to produce a similar bound for the Stokes PE. The arguments are based on the method proposed by F. Merkl and M. V. W\"{u}trich for localization of the PE of the Schr\"{o}dinger operator in a similar setting. Some additional work is needed to circumvent the complications arising from the restriction to divergence-free vector fields of the class of test functions in the variational characterization of the Stokes PE.

Abstract:
We review some old and prove some new results on the survival probability of a random walk among a Poisson system of moving traps on Z^d, which can also be interpreted as the solution of a parabolic Anderson model with a random time-dependent potential. We show that the annealed survival probability decays asymptotically as e^{-\lambda_1\sqrt{t}} for d=1, as e^{-\lambda_2 t/\log t} for d=2, and as e^{-\lambda_d t} for d>= 3, where \lambda_1 and \lambda_2 can be identified explicitly. In addition, we show that the quenched survival probability decays asymptotically as e^{-\tilde \lambda_d t}, with \tilde \lambda_d>0 for all d>= 1. A key ingredient in bounding the annealed survival probability is what is known in the physics literature as the Pascal principle, which asserts that the annealed survival probability is maximized if the random walk stays at a fixed position. A corollary of independent interest is that the expected cardinality of the range of a continuous time symmetric random walk increases under perturbation by a deterministic path.

Abstract:
An irreversible Markov-chain Monte Carlo (MCMC) algorithm with skew detailed balance conditions originally proposed by Turitsyn et al. is extended to general discrete systems on the basis of the Metropolis-Hastings scheme. To evaluate the efficiency of our proposed method, the relaxation dynamics of the slowest mode and the asymptotic variance are studied analytically in a random walk on one dimension. It is found that the performance in irreversible MCMC methods violating the detailed balance condition is improved by appropriately choosing parameters in the algorithm.

Abstract:
We derive an annealed large deviation principle for the normalised local times of a continuous-time random walk among random conductances in a finite domain in $\Z^d$ in the spirit of Donsker-Varadhan \cite{DV75}. We work in the interesting case that the conductances may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small values of the conductances and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution. As an application, we identify the logarithmic asymptotics of the lower tails of the principal eigenvalue of the randomly perturbed negative Laplace operator in the domain.

Abstract:
Let $D\subset R^d$ be a bounded domain and let $\mathcal P(D)$ denote the space of probability measures on $D$. Consider a Brownian motion in $D$ which is killed at the boundary and which, while alive, jumps instantaneously according to a spatially dependent exponential clock with intensity $\gamma V$ to a new point, according to a distribution $\mu\in\mathcal P(D)$. From its new position after the jump, the process repeats the above behavior independently of what has transpired previously. The generator of this process is an extension of the operator $-L_{\gamma,\mu}$, defined by L_{\gamma,\mu}u\equiv -\frac12\Delta u+\gamma V C_\mu(u), with the Dirichlet boundary condition, where $C_\mu$ is the "$\mu$-centering" operator defined by C_\mu(u)=u-\int_Du d\mu. The principal eigenvalue, $\lambda_0(\gamma,\mu)$, of $L_{\gamma,\mu}$ governs the exponential rate of decay of the probability of not exiting $D$ for large time. We study the asymptotic behavior of $\lambda_0(\gamma,\mu)$ as $\gamma\to\infty$. In particular, if $\mu$ possesses a density in a neighborhood of the boundary, which we call $\mu$, then \lim_{\gamma\to\infty}\gamma^{-\frac12}\lambda_0(\gamma,\mu)=\frac{\int_{\partial D}\frac\mu{\sqrt {V}}d\sigma}{\sqrt2\int_D\frac1{V}d\mu}. If $\mu$ and all its derivatives up to order $k-1$ vanish on the boundary, but the $k$-th derivative does not vanish identically on the boundary, then $\lambda_0(\gamma,\mu)$ behaves asymptotically like $c_k\gamma^{\frac{1-k}2}$, for an explicit constant $c_k$.

Abstract:
Let $D\subset R^d$ be a bounded domain and denote by $\mathcal P(D)$ the space of probability measures on $D$. Let \begin{equation*} L=\frac12\nabla\cdot a\nabla +b\nabla \end{equation*} be a second order elliptic operator. Let $\mu\in\mathcal P(D)$ and $\delta>0$. Consider a Markov process $X(t)$ in $D$ which performs diffusion in $D$ generated by the operator $\delta L$ and is stopped at the boundary, and which while running, jumps instantaneously, according to an exponential clock with spatially dependent intensity $V>0$, to a new point, according to the distribution $\mu$. The Markov process is generated by the operator $L_{\delta,\mu, V}$ defined by \begin{equation*} L_{\delta,\mu, V}\phi\equiv \delta L \phi+V(\int_D\phi d\mu-\phi). \end{equation*} %where $C_\mu$ is the % "$\mu$-centering" operator defined by %\begin{equation*} %C_\mu(\phi)=\phi-\int_D\phi d\mu. %\end{equation*} Let $\phi_{\delta,\mu,V}$ denote the solution to the Dirichlet problem \begin{equation*}\label{Dirprob} \begin{aligned} &L_{\delta,\mu,V}\phi=0\ \text{in}\ D;\\ &\phi=f\ \text{on}\ \partial D, \end{aligned} \end{equation*} where $f$ is continuous. The solution has the stochastic representation \begin{equation*} \phi_{\delta,\mu,V}(x)=E_xf(X(\tau_D)). \end{equation*} One has that $\phi_{0,\mu,V}(f)\equiv\lim_{\delta\to0}\phi_{\delta,\mu,V}(x)$ is independent of $x\in D$. We evaluate this constant in the case that $\mu$ has a density in a neighborhood of $\partial D$. We also study the asymptotic behavior as $\delta\to0$ of the principal eigenvalue $\lambda_0(\delta,\mu,V)$ for the operator $L_{\delta,\mu, V}$, which generalizes previously obtained results for the case $L=\frac12 \Delta$.

Abstract:
We study the random walk $X$ on the range of a simple random walk on $\mathbb{Z}^d$ in dimensions $d\geq 4$. When $d\geq 5$ we establish quenched and annealed scaling limits for the process $X$, which show that the intersections of the original simple random walk path are essentially unimportant. For $d=4$ our results are less precise, but we are able to show that any scaling limit for $X$ will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate that when $d=4$ similar logarithmic corrections are necessary in describing the asymptotic behaviour of the return probability of $X$ to the origin.

Abstract:
Quantum random walk in a two-dimensional lattice with randomly distributed traps is investigated. Distributions of quantum walkers are evaluated dynamically for the cases of Hadamard, Fourier, and Grover coins, and quantum to classical transition is examined as a function of the density of the traps. It is shown that traps act as a serious and additional source of quantum decoherence. Furthermore, non-trivial temporal dependence of the standard deviation of the probability distribution of the walker is found when the trapping imperfections are introduced.