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- 60-XX Probability theory and stochastic processes
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We consider the Fluctuation Dissipation Theorem (FDT) of statistical physics from a mathematical perspective. We formalize the concept of “linear response function” in the general framework of Markov processes. We show that for processes out of equilibrium it depends not only on the given Markov process X(s) but also on the chosen perturbation of it. We characterize the set of all possible response functions for a given Markov process and show that at equilibrium they all satisfy the FDT. That is,...
The Markov-Krein transform maps a positive measure on the real line to a probability measure. It is implicitly defined through an identity linking two holomorphic functions. In this paper an explicit formula is given. Its proof is obtained by considering boundary values of holomorhic functions. This transform appears in several classical questions in analysis and probability theory: Markov moment problem, Dirichlet distributions and processes, orbital measures. An asymptotic property for this transform...
In this paper, we study the Martin boundary associated with a harmonic structure given by a coupled partial differential equations system. We give an integral representation for non negative harmonic functions of this structure. In particular, we obtain such results for biharmonic functions (i.e. ) and for non negative solutions of the equation .
In this paper we treat noncoercive operators on simply connected homogeneous manifolds of negative curvature.
Martingale Hardy spaces and BMO spaces generated by an operator T are investigated. An atomic decomposition of the space is given if the operator T is predictable. We generalize the John-Nirenberg theorem, namely, we prove that the spaces generated by an operator T are all equivalent. The sharp operator is also considered and it is verified that the norm of the sharp operator is equivalent to the norm. The interpolation spaces between the Hardy and BMO spaces are identified by the real method....
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449