Levy Processes

If one chooses Pi(Vi, 0) = fi ) a non-stationary Markov process is defined, called the Wiener process or Wiener-Levy process. ) It is usually considered for f >0 alone and was originally invented for describing the stochastic behavior of the position of a Brownian particle (see VIII.3). The probability density for t > 0 is according to (2.2)... 

In the literature, this relation is commonly called the Fokker-Planck equation. It is important and instructive to point out that the derivation of the Fokker-Planck relation requires the existence of the first two moments. For the Levy processes, there does not exist a Fokker-Planck equation. 

We have tried to make plausible and physically acceptable a result of fundamental importance. In no way should the adoption of the earlier heuristic arguments diminish in the eyes of nonexpert readers the importance of these results for physics and mathematics. Levy processes seem to be ubiquitous, and it is a challenge for theoretical physicists to find a way to establish if the traditional approaches of nonequilibrium statistical mechanics can satisfactorily account for them. 

We would like to attract the attention of the reader to the case when the environment is a source of anomalous diffusion. Paz et al. [116] studied the decoherence process generated by a supra-ohmic bath, but they did not find any problem with the adoption of the decoherence theory. It is convenient to devote some attention to the case when the fluctuation E, is a source of Levy diffusion [59]. If the fluctuation E, is an uncorrelated Levy process, the characteristic function again decays exponentially, and the only significant change is that the... 

V. Seshadri and B. J. West, Fractal dimensionality of Levy processes. Proc. Natl. Acad. Sci. USA 79, 4051 (1982). 

We remark that if the jump length distance is also a Levy process, the mean-square displacement does not exist which has led to conceptual difficulties in applying this process to dielectric relaxation. Using these simplifications, one can identify two specialized forms of a continuous time random walk ... 

After addressing the Langevin and fractional Fokker-Planck formulations of Levy flight processes in some more detail, we will show that in the presence of steeper than harmonic external potentials, the situation changes drastically The forced Levy process no longer leads to an Levy stable density but instead to a multimodal PDF with steeper asymptotics than any Levy stable density. 

Strictly speaking, all naturally occurring power-laws in fractal or dynamic patterns are finite. Scale-free models nevertheless provide an efficient description of a wide variety of processes in complex systems [16,20,46,106]. This phenomenological fact is corroborated by the observation that the power-law properties of Levy processes persist strongly even in the presence of cutoffs [99]... 

Sait) symmetric a-stable Levy process Levy flight... 

This exponent corresponds to a symmetric a-stable Levy process 8 (0, a Levy flight, which is self-similar with Hurst exponent H = Xja. It follows from (3.89) that the mesoscopic density of particles is the solution to the space-fractional diffusion equation [371] ... 

The Cauchy problem involving the Riesz-Feller derivative was analyzed in [166, 260]. In the next section we discuss the general Markov random processes with independent and stationary increments, the Levy processes, for which the characteristic function is known explicitly. 

In the previous two sections we gave a brief account of the compound Poisson process and the symmetric a-stable Levy process. This section is an introduction to general one-dimensional Ldvy processes. The compound Poisson process and symmetric a-stable process are simply examples of Markov processes of Ldvy type. Readers who are interested in this topic in greater detail are referred to the books by Applebaum [15] and Sato [378]. 

Recall that a Levy process X (t) is a continuous-time stochastic process that has independent and stationary increments. It represents a natural generalization of a simple random walk defined as a sum of independent identically distributed random variables. The independence of increments ensures that Levy processes are Markov processes. The main feature of a Levy process is that it is infinitely divisible for... 

The Gamma process is an example of a Levy process with infinite number of jumps. The Gamma process Xp(t) is a pure jump Levy process with the intensity measure... 

Since T t) > 0 is a Levy process, its Laplace transform, the moment generating function can be written as... 

In this section we use the idea of subordination to obtain the space-fractional transport equation. Since r(t) is a nonnegative Levy process, the Laplace exponent l(s) defined in (3.161) can be written as... 

Note that some authors have used these operators interchangeably for the description of the mesoscopic transport process. It is clear that if L is self-adjoint, then it can be used as a transport operator and the function u(x, t) can represent the particle density. For example, the one-dimensional Brownian motion B t) has the infinitesimal generator L = 9 /9x which is self-adjoint. A symmetric a-stable Levy process on R has the generator L = 9 /9 x , which is self-adjoint too. In the next section we obtain L and L from the Chapman-Kolmogorov equation. 

---