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Harmonic functions, h-transform and large deviations for random walks in random environments in dimensions four and higher | | Atilla Yilmaz
; | | Date: |
8 Dec 2009 | | Abstract: | We consider large deviations for nearest-neighbor random walk in a uniformly
elliptic i.i.d. environment on $Z^d$. There exist variational formulae for the
quenched and the averaged rate functions $I_q$ and $I_a$, obtained by
Rosenbluth and Varadhan, respectively. $I_q$ and $I_a$ are not identically
equal. However, when $dgeq4$ and the walk satisfies the so-called (T)
condition of Sznitman, they have been previously shown to be equal on an open
set $A_{eq}$.
For every $xi$ in $A_{eq}$, we prove the existence of a positive solution to
a Laplace-like equation involving $xi$ and the original transition kernel of
the walk. We then use this solution to define a new transition kernel via the
h-transform technique of Doob. This new kernel corresponds to the unique
minimizer of Varadhan’s variational formula at $xi$. It also corresponds to
the unique minimizer of Rosenbluth’s variational formula provided that the
latter is slightly modified. | | Source: | arXiv, 0912.1429 | | Services: | Forum | Review | PDF | Favorites | |
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