Markov Processes and Magneto-Hydrodynamics Equations

Markov processes and generalized Schrödinger equations

Starting from the forward and backward infinitesimal generators of bilateral, timehomogeneous Markov processes, the self-adjoint Hamiltonians of the generalized Schrödinger equations are first introduced by means of suitable Doob transformations. Then, by broadening with the aid of the Dirichlet forms, the results of the Nelson stochastic mechanics, we prove that it is possible to associate bil...

Nonlinear Markov processes and kinetic equations

Markov processes and parabolic partial differential equations

In the first part of this article, we present the main tools and definitions of Markov processes’ theory: transition semigroups, Feller processes, infinitesimal generator, Kolmogorov’s backward and forward equations and Feller diffusion. We also give several classical examples including stochastic differential equations (SDEs) and backward SDEs (BSDEs). The second part of this article is devote...

A quaternionic structure in the three-dimensional Euler and ideal magneto-hydrodynamics equations

By considering the three-dimensional incompressible Euler equations, a 4-vector ζ is constructed out of a combination of scalar and vector products of the vorticity ω and the vortex stretching vector ω · ∇u = Sω. The evolution equation for ζ can then be cast naturally into a quaternionic Riccati equation. This is easily transformed into a quaternionic zero-eigenvalue Schrödinger equation whose ...

Sub-Fluid Models in Dissipative Magneto-Hydrodynamics