Accelerating random walks by disorder

We investigate the dynamic impact of heterogeneous environments on superdiffusive random walks known as L\'evy flights. We devote particular attention to the relative weight of source and target locations on the rates for spatial displacements of the random walk. Unlike ordinary random walks which are slowed down for all values of the relative weight of source and target, non-local superdiffusive processes show distinct regimes of attenuation and acceleration for increased source and target weight, respectively. Consequently, spatial inhomogeneities can facilitate the spread of superdiffusive processes, in contrast to common belief that external disorder generally slows down stochastic processes. Our results are based on a novel type of fractional Fokker-Planck equation which we investigate numerically and by perturbation theory for weak disorder.Comment: 8 pages, 5 figure

Accelerating Abelian Random Walks with Hyperbolic Dynamics

Given integers $d \geq 2, n \geq 1$, we consider affine random walks on torii $(\mathbb{Z} / n \mathbb{Z})^{d}$ defined as $X_{t+1} = A X_{t} + B_{t} \mod n$, where $A \in \mathrm{GL}_{d}(\mathbb{Z})$ is an invertible matrix with integer entries and $(B_{t})_{t \geq 0}$ is a sequence of iid random increments on $\mathbb{Z}^{d}$. We show that when $A$ has no eigenvalues of modulus $1$, this random walk mixes in $O(\log n \log \log n)$ steps as $n \rightarrow \infty$, and mixes actually in $O(\log n)$ steps only for almost all $n$. These results generalize those on the so-called Chung-Diaconis-Graham process, which corresponds to the case $d=1$. Our proof is based on the initial arguments of Chung, Diaconis and Graham, and relies extensively on the properties of the dynamical system $x \mapsto A^{\top} x$ on the continuous torus $\mathbb{R}^{d} / \mathbb{Z}^{d}$. Having no eigenvalue of modulus one makes this dynamical system a hyperbolic toral automorphism, a typical example of a chaotic system known to have a rich behaviour. As such our proof sheds new light on the speed-up gained by applying a deterministic map to a Markov chain.Comment: 26 page

Generalized (c,d)-entropy and aging random walks

Complex systems are often inherently non-ergodic and non-Markovian for which Shannon entropy loses its applicability. In particular accelerating, path-dependent, and aging random walks offer an intuitive picture for these non-ergodic and non-Markovian systems. It was shown that the entropy of non-ergodic systems can still be derived from three of the Shannon-Khinchin axioms, and by violating the fourth -- the so-called composition axiom. The corresponding entropy is of the form $S_{c,d} \sim \sum_i \Gamma(1+d,1-c\ln p_i)$ and depends on two system-specific scaling exponents, $c$ and $d$. This entropy contains many recently proposed entropy functionals as special cases, including Shannon and Tsallis entropy. It was shown that this entropy is relevant for a special class of non-Markovian random walks. In this work we generalize these walks to a much wider class of stochastic systems that can be characterized as `aging' systems. These are systems whose transition rates between states are path- and time-dependent. We show that for particular aging walks $S_{c,d}$ is again the correct extensive entropy. Before the central part of the paper we review the concept of $(c,d)$-entropy in a self-contained way.Comment: 8 pages, 5 eps figures. arXiv admin note: substantial text overlap with arXiv:1104.207

Accelerating Distributed Stochastic Optimization via Self-Repellent Random Walks

We study a family of distributed stochastic optimization algorithms where gradients are sampled by a token traversing a network of agents in random-walk fashion. Typically, these random-walks are chosen to be Markov chains that asymptotically sample from a desired target distribution, and play a critical role in the convergence of the optimization iterates. In this paper, we take a novel approach by replacing the standard linear Markovian token by one which follows a nonlinear Markov chain - namely the Self-Repellent Radom Walk (SRRW). Defined for any given 'base' Markov chain, the SRRW, parameterized by a positive scalar {\alpha}, is less likely to transition to states that were highly visited in the past, thus the name. In the context of MCMC sampling on a graph, a recent breakthrough in Doshi et al. (2023) shows that the SRRW achieves O(1/{\alpha}) decrease in the asymptotic variance for sampling. We propose the use of a 'generalized' version of the SRRW to drive token algorithms for distributed stochastic optimization in the form of stochastic approximation, termed SA-SRRW. We prove that the optimization iterate errors of the resulting SA-SRRW converge to zero almost surely and prove a central limit theorem, deriving the explicit form of the resulting asymptotic covariance matrix corresponding to iterate errors. This asymptotic covariance is always smaller than that of an algorithm driven by the base Markov chain and decreases at rate O(1/{\alpha}^2) - the performance benefit of using SRRW thereby amplified in the stochastic optimization context. Empirical results support our theoretical findings.Comment: Accepted for oral presentation at the Twelfth International Conference on Learning Representations (ICLR 2024

Retarding Sub- and Accelerating Super-Diffusion Governed by Distributed Order Fractional Diffusion Equations

We propose diffusion-like equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly, can not be viewed as self-affine random processes possessing a unique Hurst exponent. We prove the positivity of the solutions of the proposed equations and establish the relation to the Continuous Time Random Walk theory. We show that the distributed order time fractional diffusion equation describes the sub-diffusion random process which is subordinated to the Wiener process and whose diffusion exponent diminishes in time (retarding sub-diffusion) leading to superslow diffusion, for which the square displacement grows logarithmically in time. We also demonstrate that the distributed order space fractional diffusion equation describes super-diffusion phenomena when the diffusion exponent grows in time (accelerating super-diffusion).Comment: 11 pages, LaTe

Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: mathematical aspects

We show the asymptotic long-time equivalence of a generic power law waiting time distribution to the Mittag-Leffler waiting time distribution, characteristic for a time fractional CTRW. This asymptotic equivalence is effected by a combination of "rescaling" time and "respeeding" the relevant renewal process followed by a passage to a limit for which we need a suitable relation between the parameters of rescaling and respeeding. Turning our attention to spatially 1-D CTRWs with a generic power law jump distribution, "rescaling" space can be interpreted as a second kind of "respeeding" which then, again under a proper relation between the relevant parameters leads in the limit to the space-time fractional diffusion equation. Finally, we treat the `time fractional drift" process as a properly scaled limit of the counting number of a Mittag-Leffler renewal process.Comment: 36 pages, 3 figures (5 files eps). Invited lecture by R. Gorenflo at the 373. WE-Heraeus-Seminar on Anomalous Transport: Experimental Results and Theoretical Challenges, Physikzentrum Bad-Honnef (Germany), 12-16 July 2006; Chairmen: R. Klages, G. Radons and I.M. Sokolo