1,202 research outputs found
Liouville property, Wiener's test and unavoidable sets for Hunt processes
Let be a balayage space, , or -
equivalently - let be the set of excessive functions of a Hunt
process on a locally compact space with countable base such that separates points, every function in is the supremum of its
continuous minorants and there exist strictly positive continuous such that at infinity. We suppose that there is a Green
function for , a metric on and a decreasing function
having the doubling property such that
.
Assuming that the constant function is harmonic and balls are relatively
compact, is is shown that every positive harmonic function is constant
(Liouville property) and that Wiener's test at infinity shows, if a given set
in is unavoidable, that is, if the process hits with probability
one, wherever it starts.
An application yields that locally finite unions of pairwise disjoint balls
, , which have a certain separation property with respect to
a suitable measure on are unavoidable if and only if, for
some/any point , the series
diverges.
The results generalize and, exploiting a zero-one law for hitting
probabilities, simplify recent work by S. Gardiner and M. Ghergu, A. Mimica and
Z. Vondra\v cek, and the author
Computer Simulations of Quantum Chains
We report recent progress in computer simulations of quantum systems
described in the path-integral formulation. For the example of the
quantum chain we show that the accuracy of the simulation may greatly be
enhanced by a combination of multigrid update techniques with a refined
discretization scheme. This allows us to assess the accuracy of a variational
approximation.Comment: 5 pages, LaTeX + 2 postscript figures. Talk presented by TS at "Path
Integrals from meV to MeV: Dubna '96". See also
http://www.cond-mat.physik.uni-mainz.de/~janke/doc/home_janke.htm
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