@article{MaoutsaReichOpper2020,
  author    = {Maoutsa, Dimitra and Reich, Sebastian and Opper, Manfred},
  title     = {Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimation},
  series = {Entropy},
  volume    = {22},
  journal   = {Entropy},
  number    = {8},
  publisher = {MDPI},
  address   = {Basel},
  issn      = {1099-4300},
  doi       = {10.3390/e22080802},
  pages     = {35},
  year      = {2020},
  abstract  = {Fokker-Planck equations are extensively employed in various scientific fields as they characterise the behaviour of stochastic systems at the level of probability density functions. Although broadly used, they allow for analytical treatment only in limited settings, and often it is inevitable to resort to numerical solutions. Here, we develop a computational approach for simulating the time evolution of Fokker-Planck solutions in terms of a mean field limit of an interacting particle system. The interactions between particles are determined by the gradient of the logarithm of the particle density, approximated here by a novel statistical estimator. The performance of our method shows promising results, with more accurate and less fluctuating statistics compared to direct stochastic simulations of comparable particle number. Taken together, our framework allows for effortless and reliable particle-based simulations of Fokker-Planck equations in low and moderate dimensions. The proposed gradient-log-density estimator is also of independent interest, for example, in the context of optimal control.},
  language  = {en}
}
@article{KoltaiLiePlonka2019,
  author    = {Koltai, Peter and Lie, Han Cheng and Plonka, Martin},
  title     = {Frechet differentiable drift dependence of Perron-Frobenius and Koopman operators for non-deterministic dynamics},
  series = {Nonlinearity},
  volume    = {32},
  journal   = {Nonlinearity},
  number    = {11},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {0951-7715},
  doi       = {10.1088/1361-6544/ab1f2a},
  pages     = {4232 -- 4257},
  year      = {2019},
  abstract  = {We prove the Fr{\´e}chet differentiability with respect to the drift of Perron-Frobenius and Koopman operators associated to time-inhomogeneous ordinary stochastic differential equations. This result relies on a similar differentiability result for pathwise expectations of path functionals of the solution of the stochastic differential equation, which we establish using Girsanov's formula. We demonstrate the significance of our result in the context of dynamical systems and operator theory, by proving continuously differentiable drift dependence of the simple eigen- and singular values and the corresponding eigen- and singular functions of the stochastic Perron-Frobenius and Koopman operators.},
  language  = {en}
}
@article{CattiauxFradonKuliketal.2016,
  author    = {Cattiaux, Patrick and Fradon, Myriam and Kulik, Alexei M. and Roelly, Sylvie},
  title     = {Long time behavior of stochastic hard ball systems},
  series = {Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability},
  volume    = {22},
  journal   = {Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability},
  publisher = {International Statistical Institute},
  address   = {Voorburg},
  issn      = {1350-7265},
  doi       = {10.3150/14-BEJ672},
  pages     = {681 -- 710},
  year      = {2016},
  abstract  = {We study the long time behavior of a system of n = 2, 3 Brownian hard balls, living in R-d for d >= 2, submitted to a mutual attraction and to elastic collisions.},
  language  = {en}
}
@unpublished{GairingHoegeleKosenkova2016,
  author    = {Gairing, Jan and H{\"o}gele, Michael and Kosenkova, Tetiana},
  title     = {Transportation distances and noise sensitivity of multiplicative L{\´e}vy SDE with applications},
  volume    = {5},
  number    = {2},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-86693},
  pages     = {24},
  year      = {2016},
  abstract  = {This article assesses the distance between the laws of stochastic differential equations with multiplicative L{\´e}vy noise on path space in terms of their characteristics. The notion of transportation distance on the set of L{\´e}vy kernels introduced by Kosenkova and Kulik yields a natural and statistically tractable upper bound on the noise sensitivity. This extends recent results for the additive case in terms of coupling distances to the multiplicative case. The strength of this notion is shown in a statistical implementation for simulations and the example of a benchmark time series in paleoclimate.},
  language  = {en}
}
@book{Pilipenko2014,
  author    = {Pilipenko, Andrey},
  title     = {An introduction to stochastic differential equations with reflection},
  series = {Lectures in pure and applied mathematics},
  journal   = {Lectures in pure and applied mathematics},
  number    = {1},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  isbn      = {978-3-86956-297-1},
  issn      = {2199-4951},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-70782},
  publisher      = {Universit{\"a}t Potsdam},
  pages     = {ix, 75},
  year      = {2014},
  abstract  = {These lecture notes are intended as a short introduction to diffusion processes on a domain with a reflecting boundary for graduate students, researchers in stochastic analysis and interested readers. Specific results on stochastic differential equations with reflecting boundaries such as existence and uniqueness, continuity and Markov properties, relation to partial differential equations and submartingale problems are given. An extensive list of references to current literature is included. This book has its origins in a mini-course the author gave at the University of Potsdam and at the Technical University of Berlin in Winter 2013.},
  language  = {en}
}
@unpublished{FlandoliHoegele2014,
  author    = {Flandoli, Franco and H{\"o}gele, Michael},
  title     = {A solution selection problem with small stable perturbations},
  volume    = {3},
  number    = {8},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-71205},
  pages     = {43},
  year      = {2014},
  abstract  = {The zero-noise limit of differential equations with singular coefficients is investigated for the first time in the case when the noise is a general alpha-stable process. It is proved that extremal solutions are selected and the probability of selection is computed. Detailed analysis of the characteristic function of an exit time form on the half-line is performed, with a suitable decomposition in small and large jumps adapted to the singular drift.},
  language  = {en}
}