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

Liouville's theorem for the generalized harmonic function

In this paper, we give a more physical proof of Liouville's theorem for a class generalized harmonic functions by the method of parabolic equation

The Tur\'an problem for a family of tight linear forests

Let $\mathcal{F}$ be a family of $r$-graphs. The Tur\'an number $ex_r(n;\mathcal{F})$ is defined to be the maximum number of edges in an $r$-graph of order $n$ that is $\mathcal{F}$-free. The famous Erd\H{o}s Matching Conjecture shows that $$ex_r(n,M_{k+1}^{(r)})= \max\left\{\binom{rk+r-1}{r},\binom{n}{r}-\binom{n-k}{r}\right\},$$ where $M_{k+1}^{(r)}$ represents the $r$-graph consisting of $k+1$ disjoint edges. Motivated by this conjecture, we consider the Tur\'an problem for tight linear forests. A tight linear forest is an $r$-graph whose connected components are all tight paths or isolated vertices. Let $\mathcal{L}_{n,k}^{(r)}$ be the family of all tight linear forests of order $n$ with $k$ edges in $r$-graphs. In this paper, we prove that for sufficiently large $n$, $$ex_r(n;\mathcal{L}_{n,k}^{(r)})=\max\left\{\binom{k}{r}, \binom{n}{r}-\binom{n-\left\lfloor (k-1)/r\right \rfloor}{r}\right\}+d,$$ where $d=o(n^r)$ and if $r=3$ and $k=cn$ with $0<c<1$, if $r\geq 4$ and $k=cn$ with $0<c<1/2$. The proof is based on the weak regularity lemma for hypergraphs. We also conjecture that for arbitrary $k$ satisfying $k \equiv 1\ (mod\ r)$, the error term$d$ in the above result equals 0. We prove that the proposed conjecture implies the Erd\H{o}s Matching Conjecture directly

The Tur\'{a}n Number for Spanning Linear Forests

For a set of graphs $\mathcal{F}$, the extremal number $ex(n;\mathcal{F})$ is the maximum number of edges in a graph of order $n$ not containing any subgraph isomorphic to some graph in $\mathcal{F}$. If $\mathcal{F}$ contains a graph on $n$ vertices, then we often call the problem a spanning Tur\'{a}n problem. A linear forest is a graph whose connected components are all paths and isolated vertices. In this paper, we let $\mathcal{L}_n^k$ be the set of all linear forests of order $n$ with at least $n-k+1$ edges. We prove that when $n\geq 3k$ and $k\geq 2$, $$ex(n;\mathcal{L}_n^k)=\binom{n-k+1}{2}+ O(k^2).$$ Clearly, the result is interesting when $k=o(n)$

Discontinuous Galerkin methods and their adaptivity for the tempered fractional (convection) diffusion equations

This paper focuses on the adaptive discontinuous Galerkin (DG) methods for the tempered fractional (convection) diffusion equations. The DG schemes with interior penalty for the diffusion term and numerical flux for the convection term are used to solve the equations, and the detailed stability and convergence analyses are provided. Based on the derived posteriori error estimates, the local error indicator is designed. The theoretical results and the effectiveness of the adaptive DG methods are respectively verified and displayed by the extensive numerical experiments. The strategy of designing adaptive schemes presented in this paper works for the general PDEs with fractional operators.Comment: 31 pages, 5 figure

Galerkin Finite Element Approximations for Stochastic Space-Time Fractional Wave Equations

The traditional wave equation models wave propagation in an ideal conducting medium. For characterizing the wave propagation in inhomogeneous media with frequency dependent power-law attenuation, the space-time fractional wave equation appears; further incorporating the additive white Gaussian noise coming from many natural sources leads to the stochastic space-time fractional wave equation. This paper discusses the Galerkin finite element approximations for the stochastic space-time fractional wave equation forced by an additive space-time white noise. We firstly discretize the space-time additive noise, which introduces a modeling error and results in a regularized stochastic space-time fractional wave equation; then the regularity of the regularized equation is analyzed. For the discretization in space, the finite element approximation is used and the definition of the discrete fractional Laplacian is introduced. We derive the mean-squared $L^2$-norm priori estimates for the modeling error and for the approximation error to the solution of the regularized problem; and the numerical experiments are performed to confirm the estimates. For the time-stepping, we calculate the analytically obtained Mittag-Leffler type function.Comment: 28 page

Localization and ballistic diffusion for the tempered fractional Brownian-Langevin motion

This paper further discusses the tempered fractional Brownian motion, its ergodicity, and the derivation of the corresponding Fokker-Planck equation. Then we introduce the generalized Langevin equation with the tempered fractional Gaussian noise for a free particle, called tempered fractional Langevin equation (tfLe). While the tempered fractional Brownian motion displays localization diffusion for the long time limit and for the short time its mean squared displacement has the asymptotic form $t^{2H}$, we show that the asymptotic form of the mean squared displacement of the tfLe transits from $t^2$ (ballistic diffusion for short time) to $t^{2-2H}$, and then to $t^2$ (again ballistic diffusion for long time). On the other hand, the overdamped tfLe has the transition of the diffusion type from $t^{2-2H}$ to $t^2$ (ballistic diffusion). The tfLe with harmonic potential is also considered.Comment: 19 pages, 9 figure

Langevin dynamics for L\'evy walk with memory

Memory effects, sometimes, can not be neglected. In the framework of continuous time random walk, memory effect is modeled by the correlated waiting times. In this paper, we derive the two-point probability distribution of the stochastic process with correlated increments as well as the one of its inverse process, and present the Langevin description of L\'evy walk with memory, i.e., correlated waiting times. Based on the built Langevin picture, the properties of aging and nonstationary are discussed. The Langevin system exhibits sub-ballistic superdiffusion if the friction force is involved, while it displays super-ballistic diffusion or hyperdiffusion if there is no friction. It is discovered that the correlation of waiting times suppresses the diffusion behavior whether there is friction or not, and the stronger the correlation of waiting times becomes, the slower the diffusion is. In particular, the correlation function, correlation coefficient, ergodicity, and scaling property of the corresponding stochastic process are also investigated.Comment: 11 pages, 4 figure

L\'{e}vy-walk-like Langevin dynamics

Continuous time random walks and Langevin equations are two classes of stochastic models for describing the dynamics of particles in the natural world. While some of the processes can be conveniently characterized by both of them, more often one model has significant advantages (or has to be used) compared with the other one. In this paper, we consider the weakly damped Langevin system coupled with a new subordinator|$\alpha$-dependent subordinator with $1<\alpha<2$. We pay attention to the diffusion behaviour of the stochastic process described by this coupled Langevin system, and find the super-ballistic diffusion phenomena for the system with an unconfined potential on velocity but sub-ballistic superdiffusion phenomenon with a confined potential, which is like L\'{e}vy walk for long times. One can further note that the two-point distribution of inverse subordinator affects mean square displacement of this coupled weakly damped Langevin system in essential.Comment: 24 pages, 4 figure

Turing instability and Turing-Hopf bifurcation in diffusive Schnakenberg systems with gene expression time delay

For delayed reaction-diffusion Schnakenberg systems with Neumann boundary conditions, critical conditions for Turing instability are derived, which are necessary and sufficient. And existence conditions for Turing, Hopf and Turing-Hopf bifurcations are established. Normal forms truncated to order 3 at Turing-Hopf singularity of codimension 2, are derived. By investigating Turing-Hopf bifurcation, the parameter regions for the stability of a periodic solution, a pair of spatially inhomogeneous steady states and a pair of spatially inhomogeneous periodic solutions, are derived in $(\tau,\varepsilon)$ parameter plane ($\tau$ for time delay, $\varepsilon$ for diffusion rate). It is revealed that joint effects of diffusion and delay can lead to the occurrence of mixed spatial and temporal patterns. Moreover, it is also demonstrated that various spatially inhomogeneous patterns with different spatial frequencies can be achieved via changing the diffusion rate. And, the phenomenon that time delay may induce a failure of Turing instability observed by Gaffney and Monk (2006) are theoretically explained