Simulation of the continuous time random walk of the space-fractional diffusion equations

In this article, we discuss the solution of the space-fractional diffusion equation with and without central linear drift in the Fourier domain and show the strong connection between it and the α$\alpha$-stable Lévy distribution, 0<α<2$0<\alpha <2$. We use some relevant transformations of the independent variables x$x$ and t$t$, to find the solution of the space-fractional diffusion equation with central linear drift which is a special form of the space-fractional Fokker–Planck equation which is useful in studying the dynamic behaviour of stochastic differential equations driven by the non-Gaussian (Lévy) noises. We simulate the continuous time random walk of these models by using the Monte Carlo method.     

Tempered stable Lévy motion and transient super-diffusion

The space-fractional diffusion equation models anomalous super-diffusion. Its solutions are transition densities of a stable Lévy motion, representing the accumulation of power-law jumps. The tempered stable Lévy motion uses exponential tempering to cool these jumps. A tempered fractional diffusion equation governs the transition densities, which progress from super-diffusive early-time to diffusive late-time behavior. This article provides finite difference and particle tracking methods for solving the tempered fractional diffusion equation with drift. A temporal and spatial second-order Crank-Nicolson method is developed, based on a finite difference formula for tempered fractional derivatives. A new exponential rejection method for simulating tempered Lévy stables is presented to facilitate particle tracking codes.     

Sine-gordon type field in spacetime of arbitrary dimension. II: Stochastic quantization

Using the theory of Dirichlet forms, we prove the existence of a distribution-valued diffusion process such that the Nelson measure of a field with a bounded interaction density is its invariant probability measure. A Langevin equation in mathematically correct form is formulated which is satisfied by the process. The drift term of the equation is interpreted as a renormalized Euclidean current operator.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 2, pp. 179–197, November, 1995.     

THE SPACE—FRACTIONAL TELEGRAPH EQUATION AND THE RELATED FRACTIONAL TELEGRAPH PROCESS

The space-fractional telegraph equation is analyzed and the Fourier transform of its fundamental solution is obtained and discussed.A symmetric process with discontinuous trajectories, whose transition function satisfies the space-fractional telegraph equation, is presented. Its limiting behaviour and the connection with symmetric stable processes is also examined.     

Delay differential equations driven by Lévy processes: Stationarity and Feller properties

We consider a stochastic delay differential equation driven by a general Lévy process. Both the drift and the noise term may depend on the past, but only the drift term is assumed to be linear. We show that the segment process is eventually Feller, but in general not eventually strong Feller on the Skorokhod space. The existence of an invariant measure is shown by proving tightness of the segments using semimartingale characteristics and the Krylov–Bogoliubov method. A counterexample shows that the stationary solution in completely general situations may not be unique, but in more specific cases uniqueness is established.     

Invariant Measures for Monotone SPDEs with Multiplicative Noise Term

We study diffusion processes corresponding to infinite dimensional semilinear stochastic differential equations with local Lipschitz drift term and an arbitrary Lipschitz diffusion coefficient. We prove tightness and the Feller property of the solution to show existence of an invariant measure. As an application we discuss stochastic reaction diffusion equations.     

Finite volume and finite element methods for solving a one-dimensional space-fractional Boussinesq equation

In this paper, a new one-dimensional space-fractional Boussinesq equation is proposed. Two novel numerical methods with a nonlocal operator (using nodal basis functions) for the space-fractional Boussinesq equation are derived. These methods are based on the finite volume and finite element methods, respectively. Finally, some numerical results using fractional Boussinesq equation with the maximally positive skewness and the maximally negative skewness are given to demonstrate the strong potential of these approaches. The novel simulation techniques provide excellent tools for practical problems. These new numerical models can be extended to two- and three-dimensional fractional space-fractional Boussinesq equations in future research where we plan to apply these new numerical models for simulating the tidal water table fluctuations in a coastal aquifer.     

Maximum principles for a time–space fractional diffusion equation

In this paper, we focus on maximum principles of a time–space fractional diffusion equation. Maximum principles for classical solution and weak solution are all obtained by using properties of the time fractional derivative operator and the fractional Laplace operator. We deduce maximum principles for a full fractional diffusion equation, other than time-fractional and spatial-integer order diffusion equations.     

A parametrix construction for the fundamental solution of the evolution equation associated with a pseudo‐differential operator generating a Markov process

We use the method proposed by H. Kumano‐go in the classical case to construct a parametrix of the equation + q (x, D )u = 0 where q (x, D ) is a pseudo‐differential operator with symbol in the class introduced by W. Hoh. In case where –q (x, D ) extends to a generator of a Feller semigroup our construction yields an approximation for the transition densities of the corresponding Markov process. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A New Efficient Transform Mechanism with Convergence Analysis of the Space-Fractional Telegraph Equations

This article"s goal is to investigate the space-fractional telegraph equation using an effective method called the Adomian natural decomposition method (ANDM), which is a combination of the Adomian decomposition method (ADM) and the natural transform method (NTM). Using the Banach fixed point theorem, we explore proofs for the existence and uniqueness theorems applying it to a nonlinear differential equation. Using our method, exact solutions of the space-fractional telegraph equation and time-fractional diffusion problems have been obtained. To demonstrate the effectiveness of the suggested scheme, four examples are provided.     

Uniqueness of Solutions of the Derrida-Lebowitz-Speer-Spohn Equation and Quantum Drift-Diffusion Models

We prove uniqueness of solutions of the DLSS equation in a class of sufficiently regular functions. The global weak solutions of the DLSS equation constructed by Jüngel and Matthes belong to this class of uniqueness. We also show uniqueness of solutions for the quantum drift-diffusion equation, which contains additional drift and second-order diffusion terms. The results hold in case of periodic or Dirichlet-Neumann boundary conditions. Our proof is based on a monotonicity property of the DLSS operator and sophisticated approximation arguments; we derive a PDE satisfied by the pointwise square root of the solution, which enables us to exploit the monotonicity property of the operator.     

The space fractional diffusion equation with Feller’s operator

This paper investigates the space fractional diffusion equation with fractional Feller’s operator. The Green’s function is obtained by using Fourier transform, and the analytical solutions of some space fractional diffusion equations with initial (or initial and boundary) condition are obtained in terms of Green’s function. In addition, numerical simulations are discussed. The results indicate that the effect range of skewness parameter θ has more effect on probability density than that of parameter α. The results also explain the property of the skewness and long tail in the asymmetry diffusion process.     

Convergence of the Grünwald–Letnikov scheme for time-fractional diffusion

Using bivariate generating functions, we prove convergence of the Grünwald–Letnikov difference scheme for the fractional diffusion equation (in one space dimension) with and without central linear drift in the Fourier–Laplace domain as the space and time steps tend to zero in a well-scaled way. This implies convergence in distribution (weak convergence) of the discrete solution towards the probability of sojourn of a diffusing particle. The difference schemes allow also interpretation as discrete random walks. For fractional diffusion with central linear drift we show that in the Fourier–Laplace domain the limiting ordinary differential equation coincides with that for the solution of the corresponding diffusion equation.     

Plongement stochastique des systèmes lagrangiens

We define an operator which extends classical differentiation from smooth deterministic functions to certain stochastic processes. Based on this operator, we define a procedure which associates a stochastic analog to standard differential operators and ordinary differential equations. We call this procedure stochastic embedding. By embedding Lagrangian systems, we obtain a stochastic Euler–Lagrange equation which, in the case of natural Lagrangian systems, is called the embedded Newton equation. This equation contains the stochastic Newton equation introduced by Nelson in his dynamical theory of Brownian diffusions. Finally, we consider a diffusion with a gradient drift, a constant diffusion coefficient and having a probability density function. We prove that a necessary condition for this diffusion to solve the embedded Newton equation is that its density be the square of the modulus of a wave function solution of a linear Schrödinger equation. To cite this article: J. Cresson, S. Darses, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On reflecting boundary conditions for space-fractional equations on a finite interval: Proof of the matrix transfer technique

Even in the one-dimensional case, dealing with the analysis of space-fractional differential equations on finite domains is a difficult issue. On a finite interval coupled with zero flux boundary conditions, different approaches have been proposed to define a space-fractional differential operator and to compute the solution to the corresponding fractional problem, but to the best of our knowledge, a clear relationship between these strategies is yet to be established. Here, by using the theory of α-stable symmetric Lévy flights and the master equation, we derive a discrete representation of the non-local operator embedding in its definition the concept of reflecting boundary conditions. We refer to this discrete operator as the reflection matrix and provide (and prove) a theorem on the analytic expression of its eigenvalues and eigenvectors. We then use this result to compare the reflection matrix to the discrete operator defined via the matrix transfer technique, and establish the validity of the latter technique in producing the correct solution to a space-fractional differential equation on a finite interval with reflecting boundary conditions. We finally discuss and emphasize the challenges in the generalisation of the proposed result to more than one spatial dimension.     

On Invariant Measures for Diffusions on Banach Spaces

We consider a Banach space valued diffusion process corresponding to a stochastic evolution equation with strongly nonlinear drift. Sufficient conditions are given for the existence of a unique martingale solution and existence of an invariant measure. The resulting diffusion process is shown to be strongly Feller and irreducible. These properties yield uniqueness of invariant measure and ergodicity of the process. We also show that the invariant measure is equivalent to the invariant measure of the diffusion without drift. The main tool to show these results is the Girsanov Transformation.     

Group theoretic approach for solving the problem of diffusion of a drug through a thin membrane

The transformation group theoretic approach is applied to study the diffusion process of a drug through a skin-like membrane which tends to partially absorb the drug. Two cases are considered for the diffusion coefficient. The application of one parameter group reduces the number of independent variables by one, and consequently the partial differential equation governing the diffusion process with the boundary and initial conditions is transformed into an ordinary differential equation with the corresponding conditions. The obtained differential equation is solved numerically using the shooting method, and the results are illustrated graphically and in tables.     

Existence of solutions for fractional differential equation with p-Laplacian through variational method

In this paper, a class of fractional differential equation with p-Laplacian operator is studied based on the variational approach. Combining the mountain pass theorem with iterative technique, the existence of at least one nontrivial solution for our equation is obtained. Additionally, we demonstrate the application of our main result through an example.     

On the stochastic stability of some two-dimensional dynamical systems

We study the stochastic stability of a two-dimensional diffusion process described by a system of two nonlinear Itô stochastic differential equations. The nonlinear drift operator is assumed to be additively separated with respect to the variables, and the diffusion is assumed to be degenerate with respect to only one of the two components of the process. We consider cases in which, in the expression for the nonlinear drift vector, two of the four functions of one variable are linear. We obtain sufficient conditions for the stochastic stability of such systems. As an example, we consider a stochastic system whose deterministic part is equivalent to the classical Lienard equation.     

Asymptotic analysis of a kernel estimator for parabolic stochastic partial differential equations driven by fractional noises

We study a strongly elliptic partial differential operator with time-varying coeffcient in a parabolic diagonalizable stochastic equation driven by fractional noises. Based on the existence and uniqueness of the solution, we then obtain a kernel estimator of time-varying coeffcient and the convergence rates. An example is given to illustrate the theorem.