Analysis of random walks in dynamic random environments via L2-perturbations

We consider random walks in dynamic random environments given by Markovian dynamics on ${\mathbb{Z}}^d$. We assume that the environment has a stationary distribution $\mu$ and satisfies the Poincaré inequality w.r.t. $\mu$. The random walk is a perturbation of another random walk (called “unperturbed”). We assume that also the environment viewed from the unperturbed random walk has stationary distribution $\mu$. Both perturbed and unperturbed random walks can depend heavily on the environment and are not assumed to be finite-range. We derive a law of large numbers, an averaged invariance principle for the position of the walker and a series expansion for the asymptotic speed. We also provide a condition for non-degeneracy of the diffusion, and describe in some details equilibrium and convergence properties of the environment seen by the walker. All these results are based on a more general perturbative analysis of operators that we derive in the context of $L^2$- bounded perturbations of Markov processes by means of the so-called Dyson–Phillips expansion.     

Covariance of stochastic integrals with respect to fractional Brownian motion

We find an explicit expression for the cross-covariance between stochastic integral processes with respect to a $d$-dimensional fractional Brownian motion (fBm) $B_t$ with Hurst parameter $H>\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, where the integrands are vector fields applied to $B_t$. It provides, for example, a direct alternative proof of Y. Hu and D. Nualart’s result that the stochastic integral component in the fractional Bessel process decomposition is not itself a fractional Brownian motion.     

Representations of max-stable processes via exponential tilting

The recent contribution (Dieker and Mikosch, 2015) obtained representations of max-stable stationary Brown–Resnick process ${\zeta }_Z\left(t\right),t\in {\mathbb{R}}^d$ with spectral process $Z$ being Gaussian. With motivations from Dieker and Mikosch (2015) we derive for general $Z$, representations for ${\zeta }_Z$ via exponential tilting of $Z$. Our findings concern Dieker–Mikosch representations of max-stable processes, two-sided extensions of stationary max-stable processes, inf-argmax representation of max-stable distributions, and new formulas for generalised Pickands constants. Our applications include conditions for the stationarity of ${\zeta }_Z$, a characterisation of Gaussian distributions and an alternative proof of Kabluchko’s characterisation of Gaussian processes with stationary increments.     

Limit theory for random walks in degenerate time-dependent random environments

We study continuous-time (variable speed) random walks in random environments on ${\mathbb{Z}}^d$, $d\ge 2$, where, at time t, the walk at x jumps across edge $(x,y)$ at time-dependent rate $a_t(x,y)$. The rates, which we assume stationary and ergodic with respect to space–time shifts, are symmetric and bounded but possibly degenerate in the sense that the total jump rate from a vertex may vanish over finite intervals of time. We formulate conditions on the environment under which the law of diffusively-scaled random walk path tends to Brownian motion for almost every sample of the rates. The proofs invoke Moser iteration to prove sublinearity of the corrector in pointwise sense; a key additional input is a conversion of certain weighted energy norms to ordinary ones. Our conclusions apply to random walks on dynamical bond percolation and interacting particle systems as well as to random walks arising from the Helffer–Sjöstrand representation of gradient models with certain non-strictly convex potentials.     

Discretizing Malliavin calculus

Suppose $B$ is a Brownian motion and $B^n$ is an approximating sequence of rescaled random walks on the same probability space converging to $B$ pointwise in probability. We provide necessary and sufficient conditions for weak and strong $L^2$-convergence of a discretized Malliavin derivative, a discrete Skorokhod integral, and discrete analogues of the Clark–Ocone derivative to their continuous counterparts. Moreover, given a sequence $\left(X^n\right)$ of random variables which admit a chaos decomposition in terms of discrete multiple Wiener integrals with respect to $B^n$, we derive necessary and sufficient conditions for strong $L^2$-convergence to a $\sigma \left(B\right)$-measurable random variable $X$ via convergence of the discrete chaos coefficients of $X^n$ to the continuous chaos coefficients.     

Mesoscopic scales in hierarchical configuration models

To understand mesoscopic scaling in networks, we study the hierarchical configuration model (HCM), a random graph model with community structure. Connections between communities are formed as in a configuration model. We study the component sizes of HCM at criticality, and we study critical bond percolation. We find the conditions on the community sizes such that the critical component sizes of HCM behave similarly as in the configuration model. We show that the ordered components of a critical HCM on $N$ vertices are $O\left(N^{2∕3}\right)$. More specifically, the rescaled component sizes converge to the excursions of a Brownian motion with parabolic drift.     

Limit theorems for random walks

We consider a random walk $S_{\tau }$ which is obtained from the simple random walk $S$ by a discrete time version of Bochner’s subordination. We prove that under certain conditions on the subordinator $\tau$ appropriately scaled random walk $S_{\tau }$ converges in the Skorohod space to the symmetric $\alpha$-stable process $B^{\alpha }$. We also prove asymptotic formula for the transition function of $S_{\tau }$ similar to the Pólya’s asymptotic formula for $B^{\alpha }$.     

Asymptotics for the normalized error of the Ninomiya–Victoir scheme

In Gerbi et al. (2016) we proved strong convergence with order $1∕2$ of the Ninomiya–Victoir scheme $X^{NV,\eta }$ with time step $T∕N$ to the solution $X$ of the limiting SDE. In this paper we check that the normalized error defined by $\sqrt{N}\left(X?X^{NV,\eta }\right)$ converges to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields. The limit does not depend on the Rademacher random variables $\eta$. This result can be seen as a first step to adapt to the Ninomiya–Victoir scheme the central limit theorem of Lindeberg Feller type, derived in Ben Alaya and Kebaier (2015) for the multilevel Monte Carlo estimator based on the Euler scheme. When the Brownian vector fields commute, the limit vanishes. This suggests that the rate of convergence is greater than $1∕2$ in this case and we actually prove strong convergence with order $1$ and study the limit of the normalized error $N\left(X?X^{NV,\eta }\right)$. The limiting SDE involves the Lie brackets between the Brownian vector fields and the Stratonovich drift vector field. When all the vector fields commute, the limit vanishes, which is consistent with the fact that the Ninomiya–Victoir scheme coincides with the solution to the SDE on the discretization grid.     

Bounded stationary reflection II

Bounded stationary reflection at a cardinal λ is the assertion that every stationary subset of λ reflects but there is a stationary subset of λ that does not reflect at arbitrarily high cofinalities. We produce a variety of models in which bounded stationary reflection holds. These include models in which bounded stationary reflection holds at the successor of every singular cardinal $\mu >{\mathrm{?}}_{\omega }$ and models in which bounded stationary reflection holds at ${\mu }^{+}$ but the approachability property fails at μ.     

A second order asymptotic expansion in the local limit theorem for a simple branching random walk in Zd

Consider a branching random walk, where the underlying branching mechanism is governed by a Galton–Watson process and the migration of particles by a simple random walk in ${\mathbb{Z}}^d$. Denote by $Z_n\left(z\right)$ the number of particles of generation $n$ located at site $z\in {\mathbb{Z}}^d$. We give the second order asymptotic expansion for $Z_n\left(z\right)$. The higher order expansion can be derived by using our method here. As a by-product, we give the second order expansion for a simple random walk on ${\mathbb{Z}}^d$, which is used in the proof of the main theorem and is of independent interest.     

The free energy of the random walk pinning model

We consider the random walk pinning model. This is a random walk on ${\mathbb{Z}}^d$ whose law is given as the Gibbs measure ${\mu }_{N,Y}^{\beta }$, where the polymer measure ${\mu }_{N,Y}^{\beta }$ is defined by using the collision local time with another simple symmetric random walk $Y$ on ${\mathbb{Z}}^d$ up to time $N$. Then, at least two definitions of the phase transitions are known, described in terms of the partition function and the free energy. In this paper, we will show that the two critical points coincide and give an explicit formula for the free energy in terms of a variational representation. Also, we will prove that if $\beta$ is smaller than the critical point, then $X$ under ${\mu }_{N,Y}^{\beta }$ satisfies the central limit theorem and the invariance principle $P_Y$-almost surely.     

On the Komlós,Major and Tusnády strong approximation for some classes of random iterates

The famous results of Komlós, Major and Tusnády (see Komlós et al., 1976 [15] and Major, 1976 [17]) state that it is possible to approximate almost surely the partial sums of size $n$ of i.i.d. centered random variables in ${\mathbb{L}}^p$ ($p>2$) by a Wiener process with an error term of order $o\left(n^{1∕p}\right)$. Very recently, Berkes et al. (2014) extended this famous result to partial sums associated with functions of an i.i.d. sequence, provided a condition on a functional dependence measure in ${\mathbb{L}}_p$ is satisfied. In this paper, we adapt the method of Berkes, Liu and Wu to partial sums of functions of random iterates. Taking advantage of the Markovian setting, we shall give new dependent conditions, expressed in terms of a natural coupling (in ${\mathbb{L}}^{\infty }$ or in ${\mathbb{L}}^1$), under which the strong approximation result holds with rate $o\left(n^{1∕p}\right)$. As we shall see our conditions are well adapted to a large variety of models, including left random walks on $G{L}_d\left(\mathbb{R}\right)$, contracting iterated random functions, autoregressive Lipschitz processes, and some ergodic Markov chains. We also provide some examples showing that our ${\mathbb{L}}^1$-coupling condition is in some sense optimal.