An unified presentation of equilibrium distributions in classical and quantum statistical mechanics
On Poisson-Dirichlet problems with polynomial data
In this note we provide a probabilistic proof that Poisson and/or Dirichlet problems in an ellipsoid in R, that have polynomial data, also have polynomial solutions. Our proofs use basic stochastic calculus. The existing proofs are based on famous lemma by E. Fisher which we do not use, and present a simple martingale proof of it as well.
Occupation time sets of supports of continuous additive functionals
Probabilistic approach for comparing first eigenvalues
A geometry on the space of probabilities (II). Projective spaces and exponential families.
In this note we continue a theme taken up in part I, see [Gzyl and Recht: The geometry on the class of probabilities (I). The finite dimensional case. Rev. Mat. Iberoamericana 22 (2006), 545-558], namely to provide a geometric interpretation of exponential families as end points of geodesics of a non-metric connection in a function space. For that we characterize the space of probability densities as a projective space in the class of strictly positive functions, and these will be regarded as a...
A geometry on the space of probabilities (I). The finite dimensional case.
In this note we provide a natural way of defining exponential coordinates on the class of probabilities on the set Ω = [1,n] or on P = {p = (p, ..., p) ∈ R| p > 0; Σ p = 1}. For that we have to regard P as a projective space and the exponential coordinates will be related to geodesic flows in C.
Intrinsic geometric on the class of probability densities and exponential families.
We present a way of thinking of exponential farnilies as geodesic surfaces in the class of positive functions considered as a (multiplicative) sub-group G of the group G of all invertible elements in the algebra A of all complex bounded functions defined on a measurable space. For that we have to study a natural geometry on that algebra. The class D of densities with respect to a given rneasure will happen to be representatives of equivalence classes defining a projective space in A. The natural...
Classical analogues of quantum paradoxes.
Geometry on the space of positive functions.
On the approximation properties of Bernstein polynomials via probabilistic tools.
Recovering decay rates from noisy measurements with maximum entropy in the mean.
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