Investigations on the numerical solution of the Fokker-Planck equation with discontinuous Galerkin Methods

Nonlinear dynamic systems under stochastic excitation possess Markov characteristics. Thus, their stochastic equation of motion can be transformed into the Fokker-Planck equation which describes the evolution of the probability density. A discontinuous Galerkin (DG) method is applied to solve the Fokker-Planck equation. This method provides numerical stability as well as accuracy and is able to treat discontinuities of the solution. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the substantiation of a projection method for the stationary Fokker-Planck equation

A projection-type condition is discussed that is sufficient for the stationary Fokker-Planck equation Δu - div(u f) = 0 to be solvable within a class of probability density functions. Based on existence theorems and estimates of positive solutions obtained by the author, a fairly large class of vector fields f satisfying this condition is proposed.     

Dynamic Evasion-Interrogation Games with Uncertainty in the Context of Electromagetics

We consider two player electromagnetic evasion-pursuit games where each player must incorporate significant uncertainty into their design strategies to disguise their intension and confuse their opponent.In this paper,the evader is allowed to make dynamic changes to his strategies in response to the dynamic input with uncertainty from the interrogator.The problem is formulated in two different ways; one is based on the evolution of the probability density function of the intensity of reflected signal and leads to a controlled forward Kolmogorov or Fokker-Planck equation.The other formulation is based on the evolution of expected value of the intensity of reflected signal and leads to controlled backward Kolmogorov equations.In addition,a number of numerical results are presented to illustrate the usefulness of the proposed approach in exploring problems of control in a general dynamic game setting.     

Estimating parameters in diffusion processes using an approximate maximum likelihood approach

We present an approximate Maximum Likelihood estimator for univariate Itô stochastic differential equations driven by Brownian motion, based on numerical calculation of the likelihood function. The transition probability density of a stochastic differential equation is given by the Kolmogorov forward equation, known as the Fokker-Planck equation. This partial differential equation can only be solved analytically for a limited number of models, which is the reason for applying numerical methods based on higher order finite differences.The approximate likelihood converges to the true likelihood, both theoretically and in our simulations, implying that the estimator has many nice properties. The estimator is evaluated on simulated data from the Cox-Ingersoll-Ross model and a non-linear extension of the Chan-Karolyi-Longstaff-Sanders model. The estimates are similar to the Maximum Likelihood estimates when these can be calculated and converge to the true Maximum Likelihood estimates as the accuracy of the numerical scheme is increased. The estimator is also compared to two benchmarks; a simulation-based estimator and a Crank-Nicholson scheme applied to the Fokker-Planck equation, and the proposed estimator is still competitive.     

Dynamic Evasion-Interrogation Games with Uncertainty in the Context of Electromagnetics

We consider two player electromagnetic evasion-pursuit games where each player must incorporate significant uncertainty into their design strategies to disguise their intension and confuse their opponent. In this paper, the evader is allowed to make dynamic changes to his strategies in response to the dynamic input with uncertainty from the interrogator. The problem is formulated in two different ways. One is based on the evolution of the probability density function of the intensity of reflected signal and leads to a controlled forward Kolmogorov or Fokker-Planck equation. The other formulation is based on the evolution of expected value of the intensity of reflected signal and leads to controlled backward Kolmogorov equations. In addition, a number of numerical results are presented to illustrate the usefulness of the proposed approach in exploring problems of control in a general dynamic game setting.     

农产品价格的随机模型及风险度量

在连续时间模型的假设条件下 ,研究了农产品价格服从伊藤随机过程的数学期望及方差问题 .首先利用 Fok ker-Planck方程及偏微分方程 ,经过变形对由该扩散随机过程所描述的价格均值及风险进行了估计 ;然后给出了假设 Ito随机过程为稳态条件下的转移概率密度 ps的表达式 ,利用 ps求出相应的价格与风险值 .该模型也可用于风险投资等领域的研究 .     

On the Optimal Control of a Random Walk with Jumps and Barriers

The modeling and optimal control of a class of random walks (RWs) is investigated in the framework of the Chapman-Kolmogorov (CK) and Fokker-Planck (FP) equations. This class of RWs includes jumps driven by a compound Poisson process and are subject to different barriers. A control mechanism is investigated that is included in the CK stochastic transition matrix and the purpose of the control is to track a desired discrete probability density function and attain a desired terminal density configuration. Existence and characterization of optimal controls are discussed. The proposed approach allows the derivation of a new FP model that accommodates the presence of the jumps and guarantees conservation of total probability in the case of reflecting barriers, which are modelled by appropriate operators. Results of numerical experiments are presented that successfully validate the proposed control framework.     

Deterministic diffusion of particles

The diffusion of the particles is described in terms of a mean motion with a speed equal to the osmotic velocity associated with the diffusion process. Three numerical schemes are presented. The first two are based on the approximation of the gradient on an irregular mesh. The third is derived from a finite-element approach. Voronoi diagrams are used to handle the irregular grid of the particles. The convergence of the schemes is studied numerically, by comparing the results with the exact solution. Applications to the Fokker-Planck equation and to the problem of disposing particles according to a given probability distribution are presented.     

Quasi-invariance of Lebesgue measure under the homeomorphic flow generated by SDE with non-Lipschitz coefficient

We consider the stochastic flow generated by Stratonovich stochastic differential equations with non-Lipschitz drift coefficients. Based on the author's previous works, we show that if the generalized divergence of the drift is bounded, then the Lebesgue measure on Rd is quasi-invariant under the action of the stochastic flow, and the explicit expression of the Radon-Nikodym derivative is also presented. Finally we show in a special case that the unique solution of the corresponding Fokker-Planck equation is given by the density of the stochastic flow.     

On weak uniqueness and distributional properties of a solution to an SDE with α-stable noise

For an SDE driven by a rotationally invariant $\alpha$-stable noise we prove weak uniqueness of the solution under the balance condition $\alpha +\gamma >1$, where $\gamma$ denotes the Hölder index of the drift coefficient. We prove the existence and continuity of the transition probability density of the corresponding Markov process and give a representation of this density with an explicitly given “principal part”, and a “residual part” which possesses an upper bound. Similar representation is also provided for the derivative of the transition probability density w.r.t. the time variable.     

On the generalization of probabilistic transformation method

The probabilistic transformation method with the finite element analysis is a new technique to solve random differential equation. The advantage of this technique is finding the “exact” expression of the probability density function of the solution when the probability density function of the input is known. However the disadvantage is due to the characteristics (geometrics and materials) of the analyzed structure included in the random differential equation.

In this paper, a developed formula is used to generalize this technique by obtaining the “exact” joint probability density function of the solution in any situations, as well as the proposed technique for the non-linear case.     

On a class of probability distributions

The application of a three parameter class of one-sided probability distributions is being discussed. For specific parameter values, this class contains as special cases a number of well-known distributions of statistics and statistical physics, namely, Gauss, Weibull, exponential, Rayleigh, Gamma, chi-square, Maxwell, and Wien (limiting case of Planck's distribution). One of the three parameters represents scale; the other two represent initial and terminal shape of the associated probability density function. A fourth parameter, shift, may be introduced. The distribution class discussed in this paper was introduced by L. Amoroso [2] in 1924. It is closely connected with a family of linear Fokker-Planck equations (generalized Feller equation). In fact, the class of probability density functions associated with the distribution class considered here is a special case of the set of all delta function initial condition solutions of the generalized Feller equation for a fixed value of the time variable. It will be shown that, as a function of the logarithm of the independent variable, the logarithm of the cumulative distribution function is asymptotically linear as the independent variable approaches zero from above. This fact leads to a general criterion for the applicability of the presented distribution family relative to given empirical data. The applicability criterion can be used to determine approximate values for the two shape parameters. They can subsequently be used as initial values in any of the established parameter estimation techniques.     

调制白噪声激励下的单自由度强非线性系统的近似瞬态响应概率密度

研究调制白噪声激励下,包含弱非线性阻尼及强非线性刚度的单自由度系统的近似瞬态响应概率密度．应用基于广义谐和函数的随机平均法,导出关于幅值瞬态概率密度的平均Fokker－Planck－Kolmogorov 方程．该方程的解可近似表示为适当的正交基函数的级数和,其中系数是随时间变化的．应用Galerkin方法,这些系数可由一阶线性微分方程组解得,从而可得幅值响应的瞬态概率密度的半解析表达式及系统状态响应的瞬态概率密度和幅值的统计矩．以受调制白噪声激励的van der Pol-Duffing振子为例验证其求解过程,并讨论了线性阻尼系数及非线性刚度系数等系统参数对系统响应的影响．     

不规则星系中星体随机运动的几率密度与天体不确定关系

在这篇论文中，作者用Langevin随机微分方程来描写不规则星系中任何一个作准布朗型运动的星体的随机运动．并用相空间中的表示点（X．X）来描写其每一随机态.用Fokker－Planck方程计算出了这些随机态的稳态几率密度fs（X．X）；同时，用此fs（X．X）计算出了随机运动星体的速度和位置的方差与涨落．进而作者发现：随机运动的星体满足一种天体的不确定关系．     

Semimartingales from the Fokker-Planck Equation

We show the existence of a semimartingale of which one-dimensional marginal distributions are given by the solution of the Fokker-Planck equation with the pth integrable drift vector (p > 1).     

Stationary probability distributions of some lotka-volterra types of stochastic predation systems

This paper provides exact solutions to the stationary probability distributions in some stochastic predation systems. These are derived by solving the Fokker-Planck equations for:

(i) a generalized stochastic Lotka-Volterra predator-prey system, and

(ii) a generalised stochastic Lotka-Volterra food chain.

In all these systems the growth dynamics of all levels of species are subject to stochastic shocks. Since stationary probability distributions provide the most comprehensive characterization of a stochastic system in a steady state, system stability can be analysed accordingly     

An approximation method for a fokker-planck-vlasov equation

A simplified Fokker-Planck equation of statistical plasma physics is mathematically investigated. For the Cauchy problem a constructive approximation method is introduced. This procedure yields a sequence (fn) of approximate densities, uniformly converging to the problem's global classical solution with linear convergence velocity.     

Multi-mode model of a piezomagnetoelastic energy harvester under random excitation

Abstract: The transformation of ambient vibrational energy into electric energy through the use of piezoelectric energy harvesting devices has been the subject of numerous investigations [1]. A commonly studied energy harvesting device performing especially well under broadband excitation, is the piezomagnetoelastic energy harvester investigated by Erturk et al. [2], which is usually discretised for the fundamental vibration mode resulting in a single-mode model. This contribution presents the study of a multi-mode model of the piezomagnetoelastic energy harvester under random excitation. The probabilty density function (PDF) is computed to be the solution of the corresponding Fokker-Planck equation using a Galerkin type method [3,4]. Based on the PDF, the resulting voltage variance is computed as a measurement for the expected power output as demonstrated in [5]. The results of the multi-mode model are then compared with the results of the single-mode model. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multidimensional Limit Theorems Allowing Large Deviations for Densities of Regular Variation

The sums of i.i.d. random vectors are considered. It is assumed that the underlying distribution is absolutely continuous and its density possesses the property which can be referred to as regular variation. The asymptotic expressions for the probability of large deviations are established in the case of a normal limiting law. Furthermore, the role of the maximal summand is emphasized.     

Trajectory statistical solutions for the 3D Navier–Stokes equations: The trajectory attractor approach

In this article, we construct the trajectory statistical solution for the 3D incompressible Navier–Stokes equations via the natural translation semigroup and trajectory attractor. In our construction, the trajectory statistical solution is an invariant space–time probability measure which is carried by the trajectory attractor of the natural translation semigroup defined on the trajectory space, and the trajectory statistical solution possesses the invariant property under the acting of the translation semigroup.