Stochastic Calculus for a Time-changed Semimartingale and the Associated Stochastic Differential Equations

It is shown that under a certain condition on a semimartingale and a time-change, any stochastic integral driven by the time-changed semimartingale is a time-changed stochastic integral driven by the original semimartingale. As a direct consequence, a specialized form of the Ito formula is derived. When a standard Brownian motion is the original semimartingale, classical Ito stochastic differential equations driven by the Brownian motion with drift extend to a larger class of stochastic differential equations involving a time-change with continuous paths. A form of the general solution of linear equations in this new class is established, followed by consideration of some examples analogous to the classical equations. Through these examples, each coefficient of the stochastic differential equations in the new class is given meaning. The new feature is the coexistence of a usual drift term along with a term related to the time-change.Comment: 27 pages; typos correcte

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

Hausdorff dimension of operator semistable L\'evy processes

Let $X=\{X(t)\}_{t\geq0}$ be an operator semistable L\'evy process in \rd with exponent $E$, where $E$ is an invertible linear operator on \rd and $X$ is semi-selfsimilar with respect to $E$. By refining arguments given in Meerschaert and Xiao \cite{MX} for the special case of an operator stable (selfsimilar) L\'evy process, for an arbitrary Borel set B\subseteq\rr_+ we determine the Hausdorff dimension of the partial range $X(B)$ in terms of the real parts of the eigenvalues of $E$ and the Hausdorff dimension of $B$.Comment: 23 page

Composition of processes and related partial differential equations

In this paper different types of compositions involving independent fractional Brownian motions B^j_{H_j}(t), t>0, j=1,$ are examined. The partial differential equations governing the distributions of I_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|), t>0 and J_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|^{1/H_1}), t>0 are derived by different methods and compared with those existing in the literature and with those related to B^1(|B^2_{H_2}(t)|), t>0. The process of iterated Brownian motion I^n_F(t), t>0 is examined in detail and its moments are calculated. Furthermore for J^{n-1}_F(t)=B^1_{H}(|B^2_H(...|B^n_H(t)|^{1/H}...)|^{1/H}), t>0 the following factorization is proved J^{n-1}_F(t)=\prod_{j=1}^{n} B^j_{\frac{H}{n}}(t), t>0. A series of compositions involving Cauchy processes and fractional Brownian motions are also studied and the corresponding non-homogeneous wave equations are derived.Comment: 32 page

Subdiffusive transport in intergranular lanes on the Sun. The Leighton model revisited

In this paper we consider a random motion of magnetic bright points (MBP) associated with magnetic fields at the solar photosphere. The MBP transport in the short time range [0-20 minutes] has a subdiffusive character as the magnetic flux tends to accumulate at sinks of the flow field. Such a behavior can be rigorously described in the framework of a continuous time random walk leading to the fractional Fokker-Planck dynamics. This formalism, applied for the analysis of the solar subdiffusion of magnetic fields, generalizes the Leighton's model.Comment: 7 page

Super-diffusive Transport Processes in Porous Media

The basic assumption of models for the transport of contaminants through soil is that the movements of solute particles are characterized by the Brownian motion. However, the complexity of pore space in natural porous media makes the hypothesis of Brownian motion far too restrictive in some situations. Therefore, alternative models have been proposed. One of the models, many times encountered in hydrology, is based in fractional differential equations, which is a one-dimensional fractional advection diffusion equation where the usual second-order derivative gives place to a fractional derivative of order α, with 1 < α ≤ 2. When a fractional derivative replaces the second-order derivative in a diffusion or dispersion model, it leads to anomalous diffusion, also called super-diffusion. We derive analytical solutions for the fractional advection diffusion equation with different initial and boundary conditions. Additionally, we analyze how the fractional parameter α affects the behavior of the solutions

Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation

A detailed study is presented for a large class of uncoupled continuous-time random walks (CTRWs). The master equation is solved for the Mittag-Leffler survival probability. The properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed. Finally, some common objections found in the literature are thoroughly reviewed.Comment: Preprint version of an already published paper. 8 page

Randomly Stopped Nonlinear Fractional Birth Processes

We present and analyse the nonlinear classical pure birth process \mathpzc{N} (t), $t>0$, and the fractional pure birth process \mathpzc{N}^\nu (t), $t>0$, subordinated to various random times, namely the first-passage time $T_t$ of the standard Brownian motion $B(t)$, $t>0$, the $\alpha$-stable subordinator \mathpzc{S}^\alpha(t), $\alpha \in (0,1)$, and others. For all of them we derive the state probability distribution $\hat{p}_k (t)$, $k \geq 1$ and, in some cases, we also present the corresponding governing differential equation. We also highlight interesting interpretations for both the subordinated classical birth process \hat{\mathpzc{N}} (t), $t>0$, and its fractional counterpart \hat{\mathpzc{N}}^\nu (t), $t>0$ in terms of classical birth processes with random rates evaluated on a stretched or squashed time scale. Various types of compositions of the fractional pure birth process \mathpzc{N}^\nu(t) have been examined in the last part of the paper. In particular, the processes \mathpzc{N}^\nu(T_t), \mathpzc{N}^\nu(\mathpzc{S}^\alpha(t)), \mathpzc{N}^\nu(T_{2\nu}(t)), have been analysed, where $T_{2\nu}(t)$, $t>0$, is a process related to fractional diffusion equations. Also the related process \mathpzc{N}(\mathpzc{S}^\alpha({T_{2\nu}(t)})) is investigated and compared with \mathpzc{N}(T_{2\nu}(\mathpzc{S}^\alpha(t))) = \mathpzc{N}^\nu (\mathpzc{S}^\alpha(t)). As a byproduct of our analysis, some formulae relating Mittag--Leffler functions are obtained

Continuous-time statistics and generalized relaxation equations

Using two simple examples, the continuous-time random walk as well as a two state Markov chain, the relation between generalized anomalous relaxation equations and semi-Markov processes is illustrated. This relation is then used to discuss continuous-time random statistics in a general setting, for statistics of convolution-type. Two examples are presented in some detail: the sum statistic and the maximum statistic