Large-time behavior in non-symmetric Fokker-Planck equations

We consider three classes of linear non-symmetric Fokker-Planck equations having a unique steady state and establish exponential convergence of solutions towards the steady state with explicit (estimates of) decay rates. First, "hypocoercive" Fokker-Planck equations are degenerate parabolic equations such that the entropy method to study large-time behavior of solutions has to be modified. We review a recent modified entropy method (for non-symmetric Fokker-Planck equations with drift terms that are linear in the position variable). Second, kinetic Fokker-Planck equations with non-quadratic potentials are another example of non-symmetric Fokker-Planck equations. Their drift term is nonlinear in the position variable. In case of potentials with bounded second-order derivatives, the modified entropy method allows to prove exponential convergence of solutions to the steady state. In this application of the modified entropy method symmetric positive definite matrices solving a matrix inequality are needed. We determine all such matrices achieving the optimal decay rate in the modified entropy method. In this way we prove the optimality of previous results. Third, we discuss the spectral properties of Fokker-Planck operators perturbed with convolution operators. For the corresponding Fokker-Planck equation we show existence and uniqueness of a stationary solution. Then, exponential convergence of all solutions towards the stationary solution is proven with an uniform rate

On multi-dimensional hypocoercive BGK models

We study hypocoercivity for a class of linearized BGK models for continuous phase spaces. We develop methods for constructing entropy functionals that enable us to prove exponential relaxation to equilibrium with explicit and physically meaningful rates. In fact, we not only estimate the exponential rate, but also the second time scale governing the time one must wait before one begins to see the exponential relaxation in the L1 distance. This waiting time phenomenon, with a long plateau before the exponential decay "kicks in" when starting from initial data that is well-concentrated in phase space, is familiar from work of Aldous and Diaconis on Markov chains, but is new in our continuous phase space setting. Our strategies are based on the entropy and spectral methods, and we introduce a new "index of hypocoercivity" that is relevant to models of our type involving jump processes and not only diffusion. At the heart of our method is a decomposition technique that allows us to adapt Lyapunov's direct method to our continuous phase space setting in order to construct our entropy functionals. These are used to obtain precise information on linearized BGK models. Finally, we also prove local asymptotic stability of a nonlinear BGK model.Comment: 55 pages, 2 figure

Technical Efficiency Analysis in Male and Female-Managed Farms: A Study of Maize Production in West Pokot District, Kenya

There has been a decline in crop productivity in Kenya, a situation that has contributed to the raising food insecurity in the country. Communities in arid and semi-arid areas of the country, the landless, and female headed households are particularly vulnerable to food insecurity. Given the rising percentage of female headed households in Kenya (estimated at 37% in 2005), there is need to examine crop productivity in male and female managed farms and do a diagnosis of factors that lead to low productivity in female headed households and hence their higher vulnerability to food insecurity. This paper aims at examining gender differentials in farm resource ownership and how it affects the technical efficiency in maize production in male and female managed farms. The underlying hypothesis in this paper is that: given the same level of production technology, there should be no significant differences in the levels of maize productivity between male and female farmers. Hence, any significant differences would be attributed to differences in access to production resources. The paper focuses on maize since it is the staple crop in the study region and in Kenya. The stochastic frontier analysis reveals that both male and female managed farms are technically inefficient, since both categories of farm households produced below the production frontier. Contributing factors to the inefficiency were low levels of formal education, lack of access to credit and agriculture extension facilities, low input use and labour constraints especially in female farm households.Gender, technical efficiency, maize production, Kenya, Crop Production/Industries, Farm Management, Labor and Human Capital, Q12, Q18, Q1,

Hypocoercivity for Linear ODEs and Strong Stability for Runge--Kutta Methods

In this note, we connect two different topics from linear algebra and numerical analysis: hypocoercivity of semi-dissipative matrices and strong stability for explicit Runge--Kutta schemes. Linear autonomous ODE systems with a non-coercive matrix are called hypocoercive if they still exhibit uniform exponential decay towards the steady state. Strong stability is a property of time-integration schemes for ODEs that preserve the temporal monotonicity of the discrete solutions. It is proved that explicit Runge--Kutta schemes are strongly stable with respect to semi-dissipative, asymptotically stable matrices if the hypocoercivity index is sufficiently small compared to the order of the scheme. Otherwise, the Runge--Kutta schemes are in general not strongly stable. As a corollary, explicit Runge--Kutta schemes of order $p\in 4\N$ with $s=p$ stages turn out to be \emph{not} strongly stable. This result was proved in \cite{AAJ23}, filling a gap left open in \cite{SunShu19}. Here, we present an alternative, direct proof

On optimal decay estimates for ODEs and PDEs with modal decomposition

We consider the Goldstein-Taylor model, which is a 2-velocity BGK model, and construct the "optimal" Lyapunov functional to quantify the convergence to the unique normalized steady state. The Lyapunov functional is optimal in the sense that it yields decay estimates in $L^2$-norm with the sharp exponential decay rate and minimal multiplicative constant. The modal decomposition of the Goldstein-Taylor model leads to the study of a family of 2-dimensional ODE systems. Therefore we discuss the characterization of "optimal" Lyapunov functionals for linear ODE systems with positive stable diagonalizable matrices. We give a complete answer for optimal decay rates of 2-dimensional ODE systems, and a partial answer for higher dimensional ODE systems.Comment: 4 figure

The hypocoercivity index for the short time behavior of linear time-invariant ODE systems

We consider the class of conservative-dissipative ODE systems, which is a subclass of Lyapunov stable, linear time-invariant ODE systems. We characterize asymptotically stable, conservative-dissipative ODE systems via the hypocoercivity (theory) of their system matrices. Our main result is a concise characterization of the hypocoercivity index (an algebraic structural property of matrices with positive semi-definite Hermitian part introduced in Achleitner, Arnold, and Carlen (2018)) in terms of the short time behavior of the propagator norm for the associated conservative-dissipative ODE system