Propagation of chaos for the Burgers equation
Hirofumi Osada, Shinichi Kotani · Journal of the Mathematical Society of Japan · 1985
In [5] H. P. McKean considered systems of many particles obeying stochastic differential equations:Under the conditions of smoothness and boundedness of the coefficients, he proved that if the initial values $X_{i}^{n}(0)$ are $i.i.d$ .random variables then any fixed finite particles converge to independent copies of a one-dimensional diffusion process determined by an equationwhere $e[x, \mu]=\int_{R}e(x, y)\mu(dy)$ and $p_{t}$ is a distribution of $X(t)$ .A point of the proof is in studying the processes $\{X_{i}^{n}(t)\}$ in the infinite product probability space $\Pi_{i=1}^{\infty}\{\mu_{i}, P_{i}\}$ of identically and independently distributed initial distribution and Brownian motions, and applying Hewitt-Savage's 0-1 law to these diffusion