We present a procedure that determines the law of a random walk in an iid random environment as a function of a single "typical" trajectory. We indicate when the trajectory characterizes the law of the environment, and we say how this law can be determined. We then show how independent trajectories having the distribution of the original walk can be generated as functions of the single observed trajectory.

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A fixed 2-dimensional projection of a 3-dimensional Brownian motion is almost surely neighborhood recurrent; is this simultaneously true of all the 2-dimensional projections with probability one? Equivalently: 3-dimensional Brownian motion hits any infinite cylinder with probability one; does it hit all cylinders? This papers shows that the answer is no. Brownian motion in three dimensions avoids random cylinders and in fact avoids bodies of revolution that grow almost as fast as cones.

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