@misc{RoellyThieullen2005,
  author    = {Roelly, Sylvie and Thieullen, Mich{\`e}le},
  title     = {Duality formula for the bridges of a Brownian diffusion : application to gradient drifts},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6710},
  year      = {2005},
  abstract  = {In this paper, we consider families of time Markov fields (or reciprocal classes) which have the same bridges as a Brownian diffusion. We characterize each class as the set of solutions of an integration by parts formula on the space of continuous paths C[0; 1]; R-d) Our techniques provide a characterization of gradient diffusions by a duality formula and, in case of reversibility, a generalization of a result of Kolmogorov.},
  language  = {en}
}
@unpublished{Murr2012,
  author    = {Murr, R{\"u}diger},
  title     = {Reciprocal classes of Markov processes : an approach with duality formulae},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-63018},
  year      = {2012},
  abstract  = {In this work we are concerned with the characterization of certain classes of stochastic processes via duality formulae. First, we introduce a new formulation of a characterization of processes with independent increments, which is based on an integration by parts formula satisfied by infinitely divisible random vectors. Then we focus on the study of the reciprocal classes of Markov processes. These classes contain all stochastic processes having the same bridges, and thus similar dynamics, as a reference Markov process. We start with a resume of some existing results concerning the reciprocal classes of Brownian diffusions as solutions of duality formulae. As a new contribution, we show that the duality formula satisfied by elements of the reciprocal class of a Brownian diffusion has a physical interpretation as a stochastic Newton equation of motion. In the context of pure jump processes we derive the following new results. We will analyze the reciprocal classes of Markov counting processes and characterize them as a group of stochastic processes satisfying a duality formula. This result is applied to time-reversal of counting processes. We are able to extend some of these results to pure jump processes with different jump-sizes, in particular we are able to compare the reciprocal classes of Markov pure jump processes through a functional equation between the jump-intensities.},
  language  = {en}
}
@article{Murr2013,
  author    = {Murr, R{\"u}diger},
  title     = {Characterization of infinite divisibility by duality formulas application to Levy processes and random measures},
  series = {Stochastic processes and their application},
  volume    = {123},
  journal   = {Stochastic processes and their application},
  number    = {5},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0304-4149},
  doi       = {10.1016/j.spa.2012.12.012},
  pages     = {1729 -- 1749},
  year      = {2013},
  abstract  = {Processes with independent increments are proven to be the unique solutions of duality formulas. This result is based on a simple characterization of infinitely divisible random vectors by a functional equation in which a difference operator appears. This operator is constructed by a variational method and compared to approaches involving chaos decompositions. We also obtain a related characterization of infinitely divisible random measures.},
  language  = {en}
}