Kolmogorov equations in infinite dimensions: Well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations

We develop a new method to uniquely solve a large class of heat equations, so-called Kolmogorov equations in infinitely many variables. The equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper (Lyapunov type) functions. This way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly nonlocally Lipschitz drifts. Apart from general analytic interest, the main motivation is to apply this to uniquely solve martingale problems in the sense of Stroock--Varadhan given by stochastic partial differential equations from hydrodynamics, such as the stochastic Navier--Stokes equations. In this paper this is done in the case of the stochastic generalized Burgers equation. Uniqueness is shown in the sense of Markov flows.Comment: Published at http://dx.doi.org/10.1214/009117905000000666 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Large time behavior of weakly coupled systems of first-order Hamilton-Jacobi equations

We show a large time behavior result for class of weakly coupled systems of first-order Hamilton-Jacobi equations in the periodic setting. We use a PDE approach to extend the convergence result proved by Namah and Roquejoffre (1999) in the scalar case. Our proof is based on new comparison, existence and regularity results for systems. An interpretation of the solution of the system in terms of an optimal control problem with switching is given

Self-consistent solutions to the intersubband rate equations in quantum cascade lasers: Analysis of a GaAs/AlxGa1-xAs device

The carrier transition rates and subband populations for a GaAs/AlGaAs quantum cascade laser operating in the mid-infrared frequency range are calculated by solving the rate equations describing the electron densities in each subband self-consistently. These calculations are repeated for a range of temperatures from 20 to 300 K. The lifetime of the upper laser level found by this self-consistent method is then used to calculate the gain for this range of temperatures. At a temperature of 77 K, the gain of the laser is found to be 34 cm(-1)/(kA/cm(-2)), when only electron–longitudinal-optical phonon transitions are considered in the calculation. The calculated gain decreases to 19.6 cm(-1)/(kA/cm(-2)) when electron–electron transition rates are included, thus showing their importance in physical models of these devices. Further analysis shows that thermionic emission could be occurring in real devices. © 2001 American Institute of Physics

On Linear Differential Equations Involving a Para-Grassmann Variable

As a first step towards a theory of differential equations involving para-Grassmann variables the linear equations with constant coefficients are discussed and solutions for equations of low order are given explicitly. A connection to n-generalized Fibonacci numbers is established. Several other classes of differential equations (systems of first order, equations with variable coefficients, nonlinear equations) are also considered and the analogies or differences to the usual (''bosonic'') differential equations discussed

New derivation for the equations of motion for particles in electromagnetism

We present equations of motion for charged particles using balanced equations, and without introducing explicitly divergent quantities. This derivation contains as particular cases some well known equations of motion, as the Lorentz-Dirac equations. An study of our main equations in terms of order of the interaction with the external field conduces us to the Landau-Lifshitz equations. We find that the analysis in second order show a special behavior. We give an explicit presentation up to third order of our main equations, and expressions for the calculation of general orders.Comment: 11 pages, 2 figures. Minor changes. Closer to published versio