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We construct self-similar functions and linear operators to deduce a self-similar variant of the Laplacian operator and of the D'Alembertian wave operator. The exigence of self-similarity as a symmetry property requires the introduction of non-local particle-particle interactions. We derive a self-similar linear wave operator describing the dynamics of a quasi-continuous linear chain of infinite length with a spatially self-similar distribution of nonlocal inter-particle springs. The self-similarity of the nonlocal harmonic particle-particle interactions results in a dispersion relation of the form of a Weierstrass-Mandelbrot function which exhibits self-similar and fractal features. We also derive a continuum approximation which relates the self-similar Laplacian to fractional integrals and yields in the low-frequency regime a power law frequency-dependence of the oscillator density.

We analyze a random walk strategy on undirected regular networks involving power matrix functions of the type L α 2 where L indicates a 'simple' Laplacian matrix. We refer such walks to as 'Fractional Random Walks' with admissible interval 0 < α ≤ 2. We deduce for the Fractional Random Walk probability generating functions (network Green's functions). From these analytical results we establish a generalization of Polya's recurrence theorem for Fractional Random Walks on d-dimensional infinite lattices: The Fractional Random Walk is transient for dimensions d > α (recurrent for d ≤ α) of the lattice. As a consequence for 0 < α < 1 the Fractional Random Walk is transient for all lattice dimensions d = 1, 2, .. and in the range 1 ≤ α < 2 for dimensions d ≥ 2. Finally, for α = 2 Polya's classical recurrence theorem is recovered, namely the walk is transient only for lattice dimensions d ≥ 3. The generalization of Polya's recurrence theorem remains valid for the class of random walks with Lévy flight asymptotics for long-range steps. We also analyze for the Fractional Random Walk mean first passage probabilities, mean residence times, mean first passage times, and global mean first passage times (Kemeny constant). For the infinite 1D lattice (infinite ring) we obtain for the transient regime 0 < α < 1 closed form expressions for the fractional lattice Green's function matrix containing the escape and ever passage probabilities. The ever passage probabilities fulfill Riesz potential power law decay asymptotic behavior for nodes far from the departure node. The non-locality of the Fractional Random Walk is generated by the non-diagonality of the fractional Laplacian matrix with Lévy type heavy tailed inverse power law decay for the probability of long-range moves. This non-local and asymptotic behavior of the Fractional Random Walk introduces small world properties with emergence of Lévy flights on large (infinite) lattices.

In this paper, we explore different Markovian random walk strategies on networks with transition probabilities between nodes defined in terms of functions of the Laplacian matrix. We generalize random walk strategies with local information in the Laplacian matrix, that describes the connections of a network, to a dynamics determined by functions of this matrix. The resulting processes are non-local allowing transitions of the random walker from one node to nodes beyond its nearest neighbors. We find that only two types of Laplacian functions are admissible with distinct behaviors for long-range steps in the infinite network limit: type (i) functions generate Brownian motions, type (ii) functions Lévy flights. For this asymptotic long-range step behavior only the lowest non-vanishing order of the Laplacian function is relevant, namely first order for type (i), and fractional order for type (ii) functions.In the first part, we discuss spectral properties of the Laplacian matrix and a series of relations that are maintained by a particular type of functions that allow to define random walks on any type of undirected connected networks. Once described general properties, we explore characteristics of random walk strategies that emerge from particular cases with functions defined in terms of exponentials, logarithms and powers of the Laplacian as well as relations of these dynamics with non-local strategies like Lévy flights and fractional transport. Finally, we analyze the global capacity of these random walk strategies to explore networks like lattices and trees and different types of random and complex networks.

In this paper we investigate Kinematic Simulation (KS) consistency with the theory of Richardson [1] for two-particle diffusivity. In particular we revisit the sweeping problem. It has been argued in [2] that due to the lack of sweeping of small scales by large scales in Kinematic Simulation, the validity of Richardson's power law might be affected. Here, we argue that the discrepancies between authors on the ability of Kinematic Simulation to predict Richardson power law may be linked to the inertial subrange they have used. For small inertial subranges, KS are efficient and the significance of the sweeping can be ignored, as a result we limit the KS agreement with the Richardson scaling law t 3 for inertial subranges kN /k1 ≤ 10000. For larger inertial range KS do not fully follow the t 3 law. Unfortunately, there is no experimental data to compare KS with and draw conclusions for such large inertial subranges. It cannot be concluded either that the discrepancy between KS and Richardson's theory for larger inertial subranges is exactly taken into account by the theory developed in (Thomson & Devenish, J. Fluid Mech. 526, 2005).