58 research outputs found
Cascades of Particles Moving at Finite Velocity in Hyperbolic Spaces
A branching process of particles moving at finite velocity over the geodesic
lines of the hyperbolic space (Poincar\'e half-plane and Poincar\'e disk) is
examined. Each particle can split into two particles only once at Poisson paced
times and deviates orthogonally when splitted. At time , after
Poisson events, there are particles moving along different geodesic
lines. We are able to obtain the exact expression of the mean hyperbolic
distance of the center of mass of the cloud of particles. We derive such mean
hyperbolic distance from two different and independent ways and we study the
behavior of the relevant expression as increases and for different values
of the parameters (hyperbolic velocity of motion) and (rate of
reproduction). The mean hyperbolic distance of each moving particle is also
examined and a useful representation, as the distance of a randomly stopped
particle moving over the main geodesic line, is presented
Construction of Markov processes and associated multiplicative functionals from given harmonic measures
Let E be a noncompact locally compact second countable Hausdorff space. We consider the question when, given a family of finite nonzero measures on E that behave like harmonic measures associated with all relatively compact open sets in E (i.e. that satisfy a certain consistency condition), one can construct a Markov process on E and a multiplicative functional with values in [0, ∞) such that the hitting distributions of the process “inflated” by the multiplicative functional yield the given harmonic measures. We achieve this construction under weak continuity and local transience conditions on these measures that are natural in the theory of Markov processes, and a mild growth restriction on them. In particular, if the space E equipped with the measures satisfies the conditions of a harmonic space, such a Markov process and associated multiplicative functional exist. The result extends in a new direction the work of many authors, in probability and in axiomatic potential theory, on constructing Markov processes from given hitting distributions (i.e. from harmonic measures that have total mass no more than 1).Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47657/1/440_2005_Article_BF01192513.pd
Renormalized solutions of semilinear equations involving measure data and operator corresponding to Dirichlet form
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