@article{KellerValleriani2012,
  author    = {Keller, Peter and Valleriani, Angelo},
  title     = {Single-molecule stochastic times in a reversible bimolecular reaction},
  series = {The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr},
  volume    = {137},
  journal   = {The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr},
  number    = {8},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {0021-9606},
  doi       = {10.1063/1.4747337},
  pages     = {7},
  year      = {2012},
  abstract  = {In this work, we consider the reversible reaction between reactants of species A and B to form the product C. We consider this reaction as a prototype of many pseudobiomolecular reactions in biology, such as for instance molecular motors. We derive the exact probability density for the stochastic waiting time that a molecule of species A needs until the reaction with a molecule of species B takes place. We perform this computation taking fully into account the stochastic fluctuations in the number of molecules of species B. We show that at low numbers of participating molecules, the exact probability density differs from the exponential density derived by assuming the law of mass action. Finally, we discuss the condition of detailed balance in the exact stochastic and in the approximate treatment.},
  language  = {en}
}
@article{KellerRoellyValleriani2015,
  author    = {Keller, Peter and Roelly, Sylvie and Valleriani, Angelo},
  title     = {On time duality for Markov Chains},
  series = {Stochastic models},
  volume    = {31},
  journal   = {Stochastic models},
  number    = {1},
  publisher = {Taylor \& Francis Group},
  address   = {Philadelphia},
  issn      = {1532-6349},
  doi       = {10.1080/15326349.2014.969736},
  pages     = {98 -- 118},
  year      = {2015},
  abstract  = {For an irreducible continuous time Markov chain, we derive the distribution of the first passage time from a given state i to another given state j and the reversed passage time from j to i, each under the condition of no return to the starting point. When these two distributions are identical, we say that i and j are in time duality. We introduce a new condition called permuted balance that generalizes the concept of reversibility and provides sufficient criteria, based on the structure of the transition graph of the Markov chain. Illustrative examples are provided.},
  language  = {en}
}
@article{KellerRoellyValleriani2015,
  author    = {Keller, Peter and Roelly, Sylvie and Valleriani, Angelo},
  title     = {A Quasi Random Walk to Model a Biological Transport Process},
  series = {Methodology and computing in applied probability},
  volume    = {17},
  journal   = {Methodology and computing in applied probability},
  number    = {1},
  publisher = {Springer},
  address   = {Dordrecht},
  issn      = {1387-5841},
  doi       = {10.1007/s11009-013-9372-5},
  pages     = {125 -- 137},
  year      = {2015},
  abstract  = {Transport molecules play a crucial role for cell viability. Amongst others, linear motors transport cargos along rope-like structures from one location of the cell to another in a stochastic fashion. Thereby each step of the motor, either forwards or backwards, bridges a fixed distance and requires several biochemical transformations, which are modeled as internal states of the motor. While moving along the rope, the motor can also detach and the walk is interrupted. We give here a mathematical formalization of such dynamics as a random process which is an extension of Random Walks, to which we add an absorbing state to model the detachment of the motor from the rope. We derive particular properties of such processes that have not been available before. Our results include description of the maximal distance reached from the starting point and the position from which detachment takes place. Finally, we apply our theoretical results to a concrete established model of the transport molecule Kinesin V.},
  language  = {en}
}