29 research outputs found
Subordination Pathways to Fractional Diffusion
The uncoupled Continuous Time Random Walk (CTRW) in one space-dimension and
under power law regime is splitted into three distinct random walks: (rw_1), a
random walk along the line of natural time, happening in operational time;
(rw_2), a random walk along the line of space, happening in operational
time;(rw_3), the inversion of (rw_1), namely a random walk along the line of
operational time, happening in natural time. Via the general integral equation
of CTRW and appropriate rescaling, the transition to the diffusion limit is
carried out for each of these three random walks. Combining the limits of
(rw_1) and (rw_2) we get the method of parametric subordination for generating
particle paths, whereas combination of (rw_2) and (rw_3) yields the
subordination integral for the sojourn probability density in space-time
fractional diffusion.Comment: 20 pages, 4 figure
Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation
We present a numerical method for the Monte Carlo simulation of uncoupled
continuous-time random walks with a Levy alpha-stable distribution of jumps in
space and a Mittag-Leffler distribution of waiting times, and apply it to the
stochastic solution of the Cauchy problem for a partial differential equation
with fractional derivatives both in space and in time. The one-parameter
Mittag-Leffler function is the natural survival probability leading to
time-fractional diffusion equations. Transformation methods for Mittag-Leffler
random variables were found later than the well-known transformation method by
Chambers, Mallows, and Stuck for Levy alpha-stable random variables and so far
have not received as much attention; nor have they been used together with the
latter in spite of their mathematical relationship due to the geometric
stability of the Mittag-Leffler distribution. Combining the two methods, we
obtain an accurate approximation of space- and time-fractional diffusion
processes almost as easy and fast to compute as for standard diffusion
processes.Comment: 7 pages, 5 figures, 1 table. Presented at the Conference on Computing
in Economics and Finance in Montreal, 14-16 June 2007; at the conference
"Modelling anomalous diffusion and relaxation" in Jerusalem, 23-28 March
2008; et
Semi-Markov Graph Dynamics
In this paper, we outline a model of graph (or network) dynamics based on two
ingredients. The first ingredient is a Markov chain on the space of possible
graphs. The second ingredient is a semi-Markov counting process of renewal
type. The model consists in subordinating the Markov chain to the semi-Markov
counting process. In simple words, this means that the chain transitions occur
at random time instants called epochs. The model is quite rich and its possible
connections with algebraic geometry are briefly discussed. Moreover, for the
sake of simplicity, we focus on the space of undirected graphs with a fixed
number of nodes. However, in an example, we present an interbank market model
where it is meaningful to use directed graphs or even weighted graphs.Comment: 25 pages, 4 figures, submitted to PLoS-ON