@article{10878,
  abstract     = {Starting from a microscopic model for a system of neurons evolving in time which individually follow a stochastic integrate-and-fire type model, we study a mean-field limit of the system. Our model is described by a system of SDEs with discontinuous coefficients for the action potential of each neuron and takes into account the (random) spatial configuration of neurons allowing the interaction to depend on it. In the limit as the number of particles tends to infinity, we obtain a nonlinear Fokker-Planck type PDE in two variables, with derivatives only with respect to one variable and discontinuous coefficients. We also study strong well-posedness of the system of SDEs and prove the existence and uniqueness of a weak measure-valued solution to the PDE, obtained as the limit of the laws of the empirical measures for the system of particles.},
  author       = {Flandoli, Franco and Priola, Enrico and Zanco, Giovanni A},
  issn         = {1553-5231},
  journal      = {Discrete and Continuous Dynamical Systems},
  keywords     = {Applied Mathematics, Discrete Mathematics and Combinatorics, Analysis},
  number       = {6},
  pages        = {3037--3067},
  publisher    = {American Institute of Mathematical Sciences},
  title        = {{A mean-field model with discontinuous coefficients for neurons with spatial interaction}},
  doi          = {10.3934/dcds.2019126},
  volume       = {39},
  year         = {2019},
}

@article{1215,
  abstract     = {Two generalizations of Itô formula to infinite-dimensional spaces are given.
The first one, in Hilbert spaces, extends the classical one by taking advantage of
cancellations when they occur in examples and it is applied to the case of a group
generator. The second one, based on the previous one and a limit procedure, is an Itô
formula in a special class of Banach spaces having a product structure with the noise
in a Hilbert component; again the key point is the extension due to a cancellation. This
extension to Banach spaces and in particular the specific cancellation are motivated
by path-dependent Itô calculus.},
  author       = {Flandoli, Franco and Russo, Francesco and Zanco, Giovanni A},
  journal      = {Journal of Theoretical Probability},
  number       = {2},
  pages        = {789--826},
  publisher    = {Springer},
  title        = {{Infinite-dimensional calculus under weak spatial regularity of the processes}},
  doi          = {10.1007/s10959-016-0724-2},
  volume       = {31},
  year         = {2018},
}