---
_id: '10878'
abstract:
- lang: eng
  text: 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.
acknowledgement: "The second author has been partially supported by INdAM through
  the GNAMPA Research\r\nProject (2017) “Sistemi stocastici singolari: buona posizione
  e problemi di controllo”. The third\r\nauthor was partly funded by the Austrian
  Science Fund (FWF) project F 65."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Franco
  full_name: Flandoli, Franco
  last_name: Flandoli
- first_name: Enrico
  full_name: Priola, Enrico
  last_name: Priola
- first_name: Giovanni A
  full_name: Zanco, Giovanni A
  id: 47491882-F248-11E8-B48F-1D18A9856A87
  last_name: Zanco
citation:
  ama: Flandoli F, Priola E, Zanco GA. A mean-field model with discontinuous coefficients
    for neurons with spatial interaction. <i>Discrete and Continuous Dynamical Systems</i>.
    2019;39(6):3037-3067. doi:<a href="https://doi.org/10.3934/dcds.2019126">10.3934/dcds.2019126</a>
  apa: Flandoli, F., Priola, E., &#38; Zanco, G. A. (2019). A mean-field model with
    discontinuous coefficients for neurons with spatial interaction. <i>Discrete and
    Continuous Dynamical Systems</i>. American Institute of Mathematical Sciences.
    <a href="https://doi.org/10.3934/dcds.2019126">https://doi.org/10.3934/dcds.2019126</a>
  chicago: Flandoli, Franco, Enrico Priola, and Giovanni A Zanco. “A Mean-Field Model
    with Discontinuous Coefficients for Neurons with Spatial Interaction.” <i>Discrete
    and Continuous Dynamical Systems</i>. American Institute of Mathematical Sciences,
    2019. <a href="https://doi.org/10.3934/dcds.2019126">https://doi.org/10.3934/dcds.2019126</a>.
  ieee: F. Flandoli, E. Priola, and G. A. Zanco, “A mean-field model with discontinuous
    coefficients for neurons with spatial interaction,” <i>Discrete and Continuous
    Dynamical Systems</i>, vol. 39, no. 6. American Institute of Mathematical Sciences,
    pp. 3037–3067, 2019.
  ista: Flandoli F, Priola E, Zanco GA. 2019. A mean-field model with discontinuous
    coefficients for neurons with spatial interaction. Discrete and Continuous Dynamical
    Systems. 39(6), 3037–3067.
  mla: Flandoli, Franco, et al. “A Mean-Field Model with Discontinuous Coefficients
    for Neurons with Spatial Interaction.” <i>Discrete and Continuous Dynamical Systems</i>,
    vol. 39, no. 6, American Institute of Mathematical Sciences, 2019, pp. 3037–67,
    doi:<a href="https://doi.org/10.3934/dcds.2019126">10.3934/dcds.2019126</a>.
  short: F. Flandoli, E. Priola, G.A. Zanco, Discrete and Continuous Dynamical Systems
    39 (2019) 3037–3067.
date_created: 2022-03-18T12:33:34Z
date_published: 2019-06-01T00:00:00Z
date_updated: 2023-09-08T11:34:45Z
day: '01'
department:
- _id: JaMa
doi: 10.3934/dcds.2019126
external_id:
  arxiv:
  - '1708.04156'
  isi:
  - '000459954800003'
intvolume: '        39'
isi: 1
issue: '6'
keyword:
- Applied Mathematics
- Discrete Mathematics and Combinatorics
- Analysis
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1708.04156
month: '06'
oa: 1
oa_version: Preprint
page: 3037-3067
project:
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
  grant_number: F6504
  name: Taming Complexity in Partial Differential Systems
publication: Discrete and Continuous Dynamical Systems
publication_identifier:
  issn:
  - 1553-5231
publication_status: published
publisher: American Institute of Mathematical Sciences
quality_controlled: '1'
scopus_import: '1'
status: public
title: A mean-field model with discontinuous coefficients for neurons with spatial
  interaction
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 39
year: '2019'
...
---
_id: '1215'
abstract:
- lang: eng
  text: "Two generalizations of Itô formula to infinite-dimensional spaces are given.\r\nThe
    first one, in Hilbert spaces, extends the classical one by taking advantage of\r\ncancellations
    when they occur in examples and it is applied to the case of a group\r\ngenerator.
    The second one, based on the previous one and a limit procedure, is an Itô\r\nformula
    in a special class of Banach spaces having a product structure with the noise\r\nin
    a Hilbert component; again the key point is the extension due to a cancellation.
    This\r\nextension to Banach spaces and in particular the specific cancellation
    are motivated\r\nby path-dependent Itô calculus."
acknowledgement: Open access funding provided by Institute of Science and Technology
  (IST Austria). The second named author benefited partially from the support of the
  “FMJH Program Gaspard Monge in Optimization and Operations Research” (Project 2014-1607H).
  He is also grateful for the invitation to the Department of Mathematics of the University
  of Pisa. The third named author is grateful for the invitation to ENSTA.
article_processing_charge: Yes (via OA deal)
author:
- first_name: Franco
  full_name: Flandoli, Franco
  last_name: Flandoli
- first_name: Francesco
  full_name: Russo, Francesco
  last_name: Russo
- first_name: Giovanni A
  full_name: Zanco, Giovanni A
  id: 47491882-F248-11E8-B48F-1D18A9856A87
  last_name: Zanco
citation:
  ama: Flandoli F, Russo F, Zanco GA. Infinite-dimensional calculus under weak spatial
    regularity of the processes. <i>Journal of Theoretical Probability</i>. 2018;31(2):789-826.
    doi:<a href="https://doi.org/10.1007/s10959-016-0724-2">10.1007/s10959-016-0724-2</a>
  apa: Flandoli, F., Russo, F., &#38; Zanco, G. A. (2018). Infinite-dimensional calculus
    under weak spatial regularity of the processes. <i>Journal of Theoretical Probability</i>.
    Springer. <a href="https://doi.org/10.1007/s10959-016-0724-2">https://doi.org/10.1007/s10959-016-0724-2</a>
  chicago: Flandoli, Franco, Francesco Russo, and Giovanni A Zanco. “Infinite-Dimensional
    Calculus under Weak Spatial Regularity of the Processes.” <i>Journal of Theoretical
    Probability</i>. Springer, 2018. <a href="https://doi.org/10.1007/s10959-016-0724-2">https://doi.org/10.1007/s10959-016-0724-2</a>.
  ieee: F. Flandoli, F. Russo, and G. A. Zanco, “Infinite-dimensional calculus under
    weak spatial regularity of the processes,” <i>Journal of Theoretical Probability</i>,
    vol. 31, no. 2. Springer, pp. 789–826, 2018.
  ista: Flandoli F, Russo F, Zanco GA. 2018. Infinite-dimensional calculus under weak
    spatial regularity of the processes. Journal of Theoretical Probability. 31(2),
    789–826.
  mla: Flandoli, Franco, et al. “Infinite-Dimensional Calculus under Weak Spatial
    Regularity of the Processes.” <i>Journal of Theoretical Probability</i>, vol.
    31, no. 2, Springer, 2018, pp. 789–826, doi:<a href="https://doi.org/10.1007/s10959-016-0724-2">10.1007/s10959-016-0724-2</a>.
  short: F. Flandoli, F. Russo, G.A. Zanco, Journal of Theoretical Probability 31
    (2018) 789–826.
date_created: 2018-12-11T11:50:45Z
date_published: 2018-06-01T00:00:00Z
date_updated: 2021-01-12T06:49:09Z
day: '01'
ddc:
- '519'
department:
- _id: JaMa
doi: 10.1007/s10959-016-0724-2
file:
- access_level: open_access
  checksum: 47686d58ec21c164540f1a980ff2163f
  content_type: application/pdf
  creator: system
  date_created: 2018-12-12T10:17:13Z
  date_updated: 2020-07-14T12:44:39Z
  file_id: '5266'
  file_name: IST-2016-712-v1+1_s10959-016-0724-2.pdf
  file_size: 671125
  relation: main_file
file_date_updated: 2020-07-14T12:44:39Z
has_accepted_license: '1'
intvolume: '        31'
issue: '2'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 789-826
project:
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
  name: IST Austria Open Access Fund
publication: Journal of Theoretical Probability
publication_status: published
publisher: Springer
publist_id: '6119'
pubrep_id: '712'
quality_controlled: '1'
scopus_import: 1
status: public
title: Infinite-dimensional calculus under weak spatial regularity of the processes
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 31
year: '2018'
...