English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Motivated by considerations from neuroscience (macroscopic behavior of largeensembles of interacting neurons), we consider a population of mean fieldinteracting diffusions in $\mathbf {R}^m$ in the presence of a randomenvironment and with spatial extension: each diffusion is attached to one siteof the lattice $\mathbf {Z}^d$, and the interaction between two diffusions isattenuated by a spatial weight that depends on their positions. For a generalclass of singular weights (including the case already considered in thephysical literature when interactions obey to a power-law of parameter$0<\alpha<d$), we address the convergence as $N\to\infty$ of the empiricalmeasure of the diffusions to the solution of a deterministic McKean-Vlasovequation and prove well-posedness of this equation, even in the degenerate casewithout noise. We provide also precise estimates of the speed of thisconvergence, in terms of an appropriate weighted Wasserstein distance,exhibiting in particular nontrivial fluctuations in the power-law case when$\frac{d}{2}\leq\alpha<d$. Our framework covers the case of polynomiallybounded monotone dynamics that are especially encountered in the main models ofneural oscillators.

Mean field limit for disordered diffusions with singular interactions