Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized Burgers' equations

Jourdain, Benjamin. "Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized Burgers' equations." ESAIM: Probability and Statistics 1 (2010): 339-355. <http://eudml.org/doc/197769>.

@article{Jourdain2010,
abstract = { We prove existence and uniqueness for two classes of martingale problems involving a nonlinear but bounded drift coefficient. In the first class, this coefficient depends on the time t, the position x and the marginal of the solution at time t. In the second, it depends on t, x and p(t,x), the density of the time marginal w.r.t. Lebesgue measure. As far as the dependence on t and x is concerned, no continuity assumption is made. The results, first proved for the identity diffusion matrix, are extended to bounded, uniformly elliptic and Lipschitz continuous matrices. As an application, we show that within each class, a particular choice of the coefficients leads to a probabilistic interpretation of generalizations of Burgers' equation. },
author = {Jourdain, Benjamin},
journal = {ESAIM: Probability and Statistics},
keywords = {Nonlinear martingale problem / Burgers equation / interacting particle system / propagation of chaos.; martingale problems; identity diffusion matrix; Lipschitz continuous matrices; Burgers' equation},
language = {eng},
month = {3},
pages = {339-355},
publisher = {EDP Sciences},
title = {Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized Burgers' equations},
url = {http://eudml.org/doc/197769},
volume = {1},
year = {2010},
}

TY - JOUR
AU - Jourdain, Benjamin
TI - Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized Burgers' equations
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 1
SP - 339
EP - 355
AB - We prove existence and uniqueness for two classes of martingale problems involving a nonlinear but bounded drift coefficient. In the first class, this coefficient depends on the time t, the position x and the marginal of the solution at time t. In the second, it depends on t, x and p(t,x), the density of the time marginal w.r.t. Lebesgue measure. As far as the dependence on t and x is concerned, no continuity assumption is made. The results, first proved for the identity diffusion matrix, are extended to bounded, uniformly elliptic and Lipschitz continuous matrices. As an application, we show that within each class, a particular choice of the coefficients leads to a probabilistic interpretation of generalizations of Burgers' equation.
LA - eng
KW - Nonlinear martingale problem / Burgers equation / interacting particle system / propagation of chaos.; martingale problems; identity diffusion matrix; Lipschitz continuous matrices; Burgers' equation
UR - http://eudml.org/doc/197769
ER -