scholarly journals On the Fourier-Transformed Boltzmann Equation with Brownian Motion

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Yong-Kum Cho ◽  
Eunsil Kim
Keyword(s):  
Cauchy Problem ◽  
Brownian Motion ◽  
Boltzmann Equation ◽  
Kinetic Model ◽  
Characteristic Functions ◽  
Stability Of Solutions ◽  
Collision Dynamics ◽  
The Cauchy Problem ◽  
Global Existence Theorem ◽  
Stochastic Time

We establish a global existence theorem, and uniqueness and stability of solutions of the Cauchy problem for the Fourier-transformed Fokker-Planck-Boltzmann equation with singular Maxwellian kernel, which may be viewed as a kinetic model for the stochastic time-evolution of characteristic functions governed by Brownian motion and collision dynamics.

scholarly journals The Randomized American Option as a Classical Solution to the Penalized Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Guillaume Leduc
Keyword(s):  
Cauchy Problem ◽  
Brownian Motion ◽  
Classical Solution ◽  
Penalty Method ◽  
Infinitesimal Generator ◽  
American Option ◽  
Geometric Brownian Motion ◽  
Uniform Bound ◽  
Option Value ◽  
The Cauchy Problem

We connect the exercisability randomized American option to the penalty method by showing that the randomized American option valueuis the uniqueclassicalsolution to the Cauchy problem corresponding to thecanonicalpenalty problem for American options. We also establish a uniform bound forAu, whereAis the infinitesimal generator of a geometric Brownian motion.

scholarly journals THE EINSTEIN–BOLTZMANN SYSTEM AND POSITIVITY

2013 ◽  
Vol 10 (01) ◽  
pp. 77-104 ◽  
Author(s):  
HO LEE ◽  
ALAN D. RENDALL
Keyword(s):  
Cauchy Problem ◽  
General Relativity ◽  
Boltzmann Equation ◽  
Cross Section ◽  
Collision Cross Section ◽  
Curved Spacetime ◽  
The Boltzmann Equation ◽  
The Cauchy Problem

The Einstein–Boltzmann (EB) system is studied, with particular attention to the non-negativity of the solution of the Boltzmann equation. A new parametrization of post-collisional momenta in general relativity is introduced and then used to simplify the conditions on the collision cross-section given by Bancel and Choquet-Bruhat. The non-negativity of solutions of the Boltzmann equation on a given curved spacetime has been studied by Bichteler and Tadmon. By examining to what extent the results of these authors apply in the framework of Bancel and Choquet-Bruhat, the non-negativity problem for the EB system is resolved for a certain class of scattering kernels. It is emphasized that it is a challenge to extend the existing theory of the Cauchy problem for the EB system so as to include scattering kernels which are physically well-motivated.

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