35,964 research outputs found
Unique Continuation for Stochastic Heat Equations
We establish a unique continuation property for stochastic heat equations
evolving in a bounded domain . Our result shows that the value of the
solution can be determined uniquely by means of its value on an arbitrary open
subdomain of at any given positive time constant. Further, when is
convex and bounded, we also give a quantitative version of the unique
continuation property. As applications, we get an observability estimate for
stochastic heat equations, an approximate result and a null controllability
result for a backward stochastic heat equation
Correlation Decay up to Uniqueness in Spin Systems
We give a complete characterization of the two-state anti-ferromagnetic spin
systems which are of strong spatial mixing on general graphs. We show that a
two-state anti-ferromagnetic spin system is of strong spatial mixing on all
graphs of maximum degree at most \Delta if and only if the system has a unique
Gibbs measure on infinite regular trees of degree up to \Delta, where \Delta
can be either bounded or unbounded. As a consequence, there exists an FPTAS for
the partition function of a two-state anti-ferromagnetic spin system on graphs
of maximum degree at most \Delta when the uniqueness condition is satisfied on
infinite regular trees of degree up to \Delta. In particular, an FPTAS exists
for arbitrary graphs if the uniqueness is satisfied on all infinite regular
trees. This covers as special cases all previous algorithmic results for
two-state anti-ferromagnetic systems on general-structure graphs.
Combining with the FPRAS for two-state ferromagnetic spin systems of
Jerrum-Sinclair and Goldberg-Jerrum-Paterson, and the very recent hardness
results of Sly-Sun and independently of Galanis-Stefankovic-Vigoda, this gives
a complete classification, except at the phase transition boundary, of the
approximability of all two-state spin systems, on either degree-bounded
families of graphs or family of all graphs.Comment: 27 pages, submitted for publicatio
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