Aging Feynman-Kac Equation

Aging, the process of growing old or maturing, is one of the most widely seen natural phenomena in the world. For the stochastic processes, sometimes the influence of aging can not be ignored. For example, in this paper, by analyzing the functional distribution of the trajectories of aging particles performing anomalous diffusion, we reveal that for the fraction of the occupation time $T_+/t$ of strong aging particles, $\langle (T^+(t)^2)\rangle=\frac{1}{2}t^2$ with coefficient $\frac{1}{2}$, having no relation with the aging time $t_a$ and $\alpha$ and being completely different from the case of weak (none) aging. In fact, we first build the models governing the corresponding functional distributions, i.e., the aging forward and backward Feynman-Kac equations; the above result is one of the applications of the models. Another application of the models is to solve the asymptotic behaviors of the distribution of the first passage time, $g(t_a,t)$. The striking discovery is that for weakly aging systems, $g(t_a,t)\sim t_a^{\frac{\alpha}{2}}t^{-1-\frac{\alpha}{2}}$, while for strongly aging systems, $g(t_a,t)$ behaves as $t_a^{\alpha-1}t^{-\alpha}$.Comment: 13 pages, 7 figure

Second order WSGD operators II: A new family of difference schemes for space fractional advection diffusion equation

The second order weighted and shifted Gr\"{u}nwald difference (WSGD) operators are developed in [Tian et al., arXiv:1201.5949] to solve space fractional partial differential equations. Along this direction, we further design a new family of second order WSGD operators; by properly choosing the weighted parameters, they can be effectively used to discretize space (Riemann-Liouville) fractional derivatives. Based on the new second order WSGD operators, we derive a family of difference schemes for the space fractional advection diffusion equation. By von Neumann stability analysis, it is proved that the obtained schemes are unconditionally stable. Finally, extensive numerical experiments are performed to demonstrate the performance of the schemes and confirm the convergent orders.Comment: 21 pages, 5 figure

High order algorithm for the time-tempered fractional Feynman-Kac equation

We provide and analyze the high order algorithms for the model describing the functional distributions of particles performing anomalous motion with power-law jump length and tempered power-law waiting time. The model is derived in [Wu, Deng, and Barkai, Phys. Rev. E., 84 (2016), 032151], being called the time-tempered fractional Feynman-Kac equation. The key step of designing the algorithms is to discretize the time tempered fractional substantial derivative, being defined as $${^S\!}D_t^{\gamma,\widetilde{\lambda}} G(x,p,t)\!=\!D_t^{\gamma,\widetilde{\lambda}} G(x,p,t)\!-\!\lambda^\gamma G(x,p,t) ~{\rm with}~\widetilde{\lambda}=\lambda+ pU(x),\, p=\rho+J\eta,\, J=\sqrt{-1},$$ where $$D_t^{\gamma,\widetilde{\lambda}} G(x,p,t) =\frac{1}{\Gamma(1-\gamma)} \left[\frac{\partial}{\partial t}+\widetilde{\lambda} \right] \int_{0}^t{\left(t-z\right)^{-\gamma}}e^{-\widetilde{\lambda}\cdot(t-z)}{G(x,p,z)}dz,$$ and $\lambda \ge 0$, $00$, and $\eta$ is a real number. The designed schemes are unconditionally stable and have the global truncation error $\mathcal{O}(\tau^2+h^2)$, being theoretically proved and numerically verified in {\em complex} space. Moreover, some simulations for the distributions of the first passage time are performed, and the second order convergence is also obtained for solving the `physical' equation (without artificial source term).Comment: 21 pages, 4 figure

Feynman-Kac Equations for Reaction and Diffusion Processes

This paper provides a theoretical framework of deriving the forward and backward Feynman-Kac equations for the distribution of functionals of the path of a particle undergoing both diffusion and chemical reaction. Very general forms of the equations are obtained. Once given the diffusion type and reaction rate, a specific forward or backward Feynman-Kac equation can be obtained. The listed in the paper include the ones for normal/anomalous diffusions and reactions with linear/nonlinear rates. Using the derived equations, we also study the occupation time in half-space, the first passage time to a fixed boundary, and the occupation time in half-space with absorbing or reflecting boundary conditions.Comment: 15 pages, 4 figure

Convergence proof for the multigrid method of the nonlocal model

Recently, nonlocal models attract the wide interests of scientist. They mainly come from two applied scientific fields: peridyanmics and anomalous diffusion. Even though the matrices of the algebraic equation corresponding the nonlocal models are usually Toeplitz (denote a0 as the principal diagonal element, a1 as the trailing diagonal element, etc). There are still some differences for the models in these two fields. For the model of anomalous diffusion, a0/a1 is uniformly bounded; most of the time, a0/a1 of the model for peridyanmics is unbounded as the stepsize h tends to zero. Based on the uniform boundedness of a0/a1, the convergence of the two-grid method is well established [Chan, Chang, and Sun, SIAM J. Sci. Comput., 19 (1998), pp. 516--529; Pang and Sun, J. Comput. Phys., 231 (2012), pp. 693--703; Chen, Wang, Cheng, and Deng, BIT, 54 (2014), pp. 623--647]. This paper provides the detailed proof of the convergence of the two-grid method for the nonlocal model of peridynamics. Some special cases of the full multigrid and the V-cycle multigrid are also discussed. The numerical experiments are performed to verify the convergence.Comment: 21 page

First exit and Dirichlet problem for the nonisotropic tempered α\alpha-stable processes

This paper discusses the first exit and Dirichlet problems of the nonisotropic tempered $\alpha$-stable process $X_t$. The upper bounds of all moments of the first exit position $\left|X_{\tau_D}\right|$ and the first exit time $\tau_D$ are firstly obtained. It is found that the probability density function of $\left|X_{\tau_D}\right|$ or $\tau_D$ exponentially decays with the increase of $\left|X_{\tau_D}\right|$ or $\tau_D$, and $\mathrm{E}\left[\tau_D\right]\sim \left|\mathrm{E}\left[X_{\tau_D}\right]\right|$,\ $\mathrm{E}\left[\tau_D\right]\sim\mathrm{E}\left[\left|X_{\tau_D}-\mathrm{E}\left[X_{\tau_D}\right]\right|^2\right]$. Since $\mathrm{\Delta}^{\alpha/2,\lambda}_m$ is the infinitesimal generator of the anisotropic tempered stable process, we obtain the Feynman-Kac representation of the Dirichlet problem with the operator $\mathrm{\Delta}^{\alpha/2,\lambda}_m$. Therefore, averaging the generated trajectories of the stochastic process leads to the solution of the Dirichlet problem, which is also verified by numerical experiments.Comment: 23 pages, 5 figure

WSLD operators II: the new fourth order difference approximations for space Riemann-Liouville derivative

High order discretization schemes play more important role in fractional operators than classical ones. This is because usually for classical derivatives the stencil for high order discretization schemes is wider than low order ones; but for fractional operators the stencils for high order schemes and low order ones are the same. Then using high order schemes to solve fractional equations leads to almost the same computational cost with first order schemes but the accuracy is greatly improved. Using the fractional linear multistep methods, Lubich obtains the $\nu$-th order ($\nu\leq 6$) approximations of the $\alpha$-th derivative ($\alpha>0$) or integral ($\alpha<0) [Lubich, SIAM J. Math. Anal., 17, 704-719, 1986], because of the stability issue the obtained scheme can not be directly applied to the space fractional operator with$\alpha \in(1,2)$ for time dependent problem. By weighting and shifting Lubich's 2nd order discretization scheme, in [Chen & Deng, arXiv:1304.7425] we derive a series of effective high order discretizations for space fractional derivative, called WSLD opeartors there. As the sequel of the previous work, we further provide new high order schemes for space fractional derivatives by weighting and shifting Lubich's 3rd and 4th order discretizations. In particular, we prove that the obtained 4th order approximations are effective for space fractional derivatives. And the corresponding schemes are used to solve the space fractional diffusion equation with variable coefficients.Comment: 22 pages, 2 figure

Multiresolution Galerkin method for solving the functional distribution of anomalous diffusion described by time-space fractional diffusion equation

The functional distributions of particle trajectories have wide applications, including the occupation time in half-space, the first passage time, and the maximal displacement, etc. The models discussed in this paper are for characterizing the distribution of the functionals of the paths of anomalous diffusion described by time-space fractional diffusion equation. This paper focuses on providing effective computation methods for the models. Two kinds of time stepping schemes are proposed for the fractional substantial derivative. The multiresolution Galerkin method with wavelet B-spline is used for space approximation. Compared with the finite element or spectral polynomial bases, the wavelet B-spline bases have the advantage of keeping the Toeplitz structure of the stiffness matrix, and being easy to generate the matrix elements and to perform preconditioning. The unconditional stability and convergence of the provided schemes are theoretically proved and numerically verified. Finally, we also discuss the efficient implementations and some extensions of the schemes, such as the wavelet preconditioning and the non-uniform time discretization.Comment: 31 pages, 1 figur

High order schemes for the tempered fractional diffusion equations

L\'{e}vy flight models whose jumps have infinite moments are mathematically used to describe the superdiffusion in complex systems. Exponentially tempering the Levy measure of L\'{e}vy flights leads to the tempered stable L\'{e}vy processes which combine both the $\alpha$-stable and Gaussian trends; and the very large jumps are unlikely and all their moments exist. The probability density functions of the tempered stable L\'{e}vy processes solve the tempered fractional diffusion equation. This paper focuses on designing the high order difference schemes for the tempered fractional diffusion equation on bounded domain. The high order difference approximations, called the tempered and weighted and shifted Gr\"{u}nwald difference (tempered-WSGD) operators, in space are obtained by using the properties of the tempered fractional calculus and weighting and shifting their first order Gr\"{u}nwald type difference approximations. And the Crank-Nicolson discretization is used in the time direction. The stability and convergence of the presented numerical schemes are established; and the numerical experiments are performed to confirm the theoretical results and testify the effectiveness of the schemes.Comment: 23 pages, 3 figure

Applications of Wavelet Bases to The Numerical Solutions of Fractional PDEs

For describing the probability distribution of the positions and times of particles performing anomalous motion, fractional PDEs are derived from the continuous time random walk models with waiting time distribution having divergent first order moment and/or jump length distribution which has divergent second order moment. It can be noted that the fractional PDEs are essentially dealing with the multiscale issues. Generally the regularity of the solutions for fractional PDEs is weak at the areas close to boundary and initial time. This paper focuses on developing the applications of wavelet bases to numerically solving fractional PDEs and digging out the potential benefits of wavelet methods comparing with other numerical methods, especially in the aspects of realizing preconditioning, adaptivity, and keeping the Toeplitz structure. More specifically, the contributions of this paper are as follows: 1. the techniques of efficiently generating stiffness matrix with computational cost $\mathcal{O}(2^J)$ are provided for first, second, and any order bases; 2. theoretically and numerically discuss the effective preconditioner for time-independent equation and multigrid method for time-dependent equation, respectively; 3. the wavelet adaptivity is detailedly discussed and numerically applied to solving the time-dependent (independent) equations. In fact, having reliable, simple, and local regularity indicators is the striking benefit of the wavelet in adaptively solving fractional PDEs (it seems hard to give a local posteriori error estimate for the adaptive finite element method because of the global property of the operator).Comment: 38 page