Density estimates for a random noise propagating through a chain of differential equations

作者：Delarue Francois^*; Menozzi Stephane

来源：Journal of Functional Analysis, 2010, 259(6): 1577-1630.

DOI：10.1016/j.jfa.2010.05.002

• Aronson estimates
• Gaussian bounds
• Hypoellipticity
• Hormander conditions
• Stochastic control

We here provide two sided bounds for the density of the solution of a system of 11 differential equations of dimension d, the first one being forced by a non-degenerate random noise and the n - 1 other ones being degenerate. The system formed by the n equations satisfies a suitable Hormander condition: the second equation feels the noise plugged into the first equation, the third equation feels the noise transmitted from the first to the second equation and so on ..., so that the noise propagates one way through the system. When the coefficients of the system are Lipschitz continuous, we show that the density of the solution satisfies Gaussian bounds with non-diffusive time scales. The proof relies on the interpretation of the density of the solution as the value function of some optimal stochastic control problem.

• 出版日期2010-9-15