I am an undergraduate math student preparing my thesis. Currently I am reading L.D Brown's (1971) paper Admissible Estimators, Recurrent Diffusions, and Insoluble Boundary Value Problems. Here is a link of the paper http://www.stat.yale.edu/~hz68/619/Brown1971.pdf. One of the main ideas of the paper is to transfer the necessary and sufficient condition for the admissibility of an estimator to the recurrence of the "corresponding" diffusions. I have some difficulty of understanding what "corresponding" means in this case.

On page 862, (1.3.10), this says if we write $\sum j''_{ii}+\sum\frac{{f'_i}^*}{f^*}j_i'=0$, then left-hand side of is actually the generator of the diffusion with local variance matrix $2I$ and local mean $\nabla f^*/f^*$

I would like to know that why this operator generates this particular diffusion, since Brown didn't say this explicitly in his paper.

I also want to know the reasoning of the statement in the following paragraph, which says if we choose $\delta$ as the usual estimator $x$, then the diffusion corresponding is a version of Brownian motion.

I read http://stats.stackexchange.com/questions/13494/intuition-behind-why-steins-paradox-only-applies-in-dimensions-ge-3 which arose my interest in reading this paper.

I will aprriciate it if someone can explain this to me.