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Suppose you have the simple random walk on $L=\mathbb{Z}^n,$ and $A \subset L$ is a subset (in my case, the subset is a union of hyperplanes, so in particular infinite). There is a random variable $T(A)$ -- the expected time to get from the origin to $A.$ The question is: is there some standard technology to upper bound the expectation of $T(A)?$ Any particularly recommended things to read?

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Perhaps this paper will provide ideas: arxiv.org/abs/1206.6568 . "Lyapunov exponents of random walks in small random potential: the lower bound" by Mountford and Mourrat. They compute, under various conditions, "the first time at which the random walk [in $\mathbb{Z}^d$] crosses the hyperplane orthogonal to $\ell$ lying at distance $n$ from the origin." –  Joseph O'Rourke Sep 3 '12 at 16:58
    
Thanks! I will check it out... –  Igor Rivin Sep 3 '12 at 17:38
    
There is a book by Sydney Redner: A guide to first-passage processes, Cambridge UP, 2001. It rather discusses physical processes, and continuous random walks, often with n<=3, but may help to find bounds also for the discrete case and general n. –  Karl Fabian Sep 3 '12 at 20:36
    
@Karl: thanks, I will try to look it up... –  Igor Rivin Sep 3 '12 at 22:32
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