Fractional governing equations for coupled random walks

Langevin Picture of Lévy Walks and Their Extensions

“…The above definition of N(t) assures that the walker starts its motion at the origin and waits there T 1 long before making its first jump. This is the so-called wait-jump scenario, which plays a crucial role in the derivation of scaling limits of coupled CTRWs, [2,[22][23][24].…”

Limit theorems and governing equations for Lévy walks

“…f (x, g, x + z, g + w) φ(dz, dw) dg dx, (5.3) which in somewhat more intuitive notation reads P ((X t , G t , Y t , H t ) ∈ (dx, dg, dy, dh)) = 1(g < t < h)u(x, g)φ(dy − x, dh − g) dx dg It is now easy to calculate the Fourier-Laplace transform of the law of L t , at first with θ held fixed (or rather λ concentrated at θ): Now recall from [36] and [23] that, given a weakly measurable family h(dx, t) (t ≥ 0) of bounded measures on R m , we say that a family ρ t (dx) of probability measures is the mild solution to the pseudo-differential equation ψ(i∇ x , ∂ t )ρ t (dx) = h(dx, t) if its Fourier-Laplace transform (FLT) solves the corresponding algebraic equation. In this sense, we can prove: Theorem 5.6.…”

“…As in [36] and [23], we consider pseudo-differential equations in the variables x and t, and solutions are defined in the "mild" sense. Our first step is to express the law of the Lévy Walk Limit in terms of its "model parameters" β ∈ (0, 1) and λ.…”

“…There is an extensive literature in this field: general results for the scaling limits of CTRWs on the stochastic process level can be found in [6,7,29,35,37,44,45]. Governing equations for the densities of the CTRW limits and the related fractional Cauchy problems were analyzed in [2,4,20,23,33,37,45]. Some recent results for particular classes of correlated and coupled CTRWs can be found in [17,24,30,31,47] The trajectories of CTRW are step functions, thus they are discontinuous.…”

Asymptotic densities of ballistic Lévy walks

“…In the wait-first model the particle jumps at the end of the waiting time, whereas in the jump-first model the particle jumps and then rests for the corresponding waiting time. Such random walks were previously considered in the literature [18][19][20][21][22][23][24][25][26], and recently also with a method of infinite densities [27] in the sub-ballistic superdiffusive regime (flight or waiting times with finite mean, 1 < γ < 2). However, in general the exact analytical solutions describing the density of random walking particles are rare and can be obtained only for some particular values of γ [16,28].…”