Strong solutions of semilinear SPDEs with unbounded diffusion

Bechtold, Florian

doi:10.48550/arxiv.2001.08848

Cited by 1 publication

(1 citation statement)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We start by observing that the noise term w n t (h) in (3.2) is neither a stochastic convolution that could be treated using a maximal inequality in Hilbert spaces (e.g. [7, §6.4] and [2] in the context of an unbounded diffusion operator), nor a classical controlled rough path integral (e.g. [15]) as the integrand depends on the upper integration limit.…”

Section: 2mentioning

confidence: 99%

A Law of Large Numbers for interacting diffusions via a mild formulation

Bechtold,

Coppini

2020

Preprint

Self Cite

View full text Add to dashboard Cite

Consider a system of n weakly interacting particles driven by independent Brownian motions. In many instances, it is well known that the empirical measure converges to the solution of a partial differential equation, usually called McKean-Vlasov or Fokker-Planck equation, as n tends to infinity. We propose a relatively new approach to show this convergence by directly studying the stochastic partial differential equation that the empirical measure satisfies for each fixed n. Under a suitable control on the noise term, which appears due to the finiteness of the system, we are able to prove that the stochastic perturbation goes to zero, showing that the limiting measure is a solution to the classical McKean-Vlasov equation. In contrast with known results, we do not require any independence or finite moment assumption on the initial condition, but the only weak convergence.The evolution of the empirical measure is studied in a suitable class of Hilbert spaces where the noise term is controlled using two distinct but complementary techniques: rough paths theory and maximal inequalities for self-normalized processes.

show abstract

Section: 2mentioning

confidence: 99%