Probability seminar: Some McKean--Vlasov problems on the half-line

6 November 2015, 3.40 PM - 6 November 2015, 4.40 PM

Sean Ledger, Bristol

SM3, School of Mathematics

Sean Ledger, Bristol

Some McKean--Vlasov problems on the half-line

We present two models of interacting continuous-time particle systems on the half-line and study their large population limits. One is motivated from mathematical finance and the other by mathematical neuroscience.

In the first, we consider a population of diffusions interacting through a correlation that is a function of the proportion of particles that have hit an absorbing threshold. It is shown that the system converges to the unique solution of a non-linear heat equation with random transport and discontinuous coefficients. A useful tool here is the Skorokhod M1 topology on the space of cadlag processes taking values in the tempered distributions.

For the second example, we consider a contagious system where each particle receives a kick towards the absorbing boundary whenever another particle reaches that boundary. The large population limit gives rise to the corresponding PDE and M--V problems. It is shown that solutions must blow-up in finite time if this positive feedback effect is made too strong. From a practical perspective this is desirable, so we consider how to adjust the notion of a solution to accommodate this effect. 

Contact information

Organisers: Marton Balazs, Haeran Cho

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