Well-Posedness and stability of Stochastic Evolution Equations arising from Neuroscience

02.06.2017, 10.15-11.45  –  2.09.0.13
Forschungsseminar Wahrscheinlichkeitstheorie

Jennifer Krüger (Berlin)

We study the existence and uniqueness of mild solutions to the deterministic and the stochastic neural field equation
with Heaviside firing rate. Since standard well-posedness results do not apply in case of a discontinuous firing rate,
we present a monotone Picard iteration scheme to show the existence of a maximal mild solution. Further, we
illustrate that general uniqueness does not hold and therefore investigate uniqueness under suitable additional
properties of the solutions. Here a novel criterion, the so-called absolute-continuity condition is introduced.
Moreover, we observe regularisation by noise: With a suitable choice of spatially correlated additive noise
uniqueness is restored without imposing any additional structural assumptions. In the second part of the talk we
present a multiscale analysis of 1D stochastic bistable reaction-diffusion equations with additive noise. It is shown
with explicit error estimates on appropriate function spaces that up to lower order w.r.t. the noise amplitude, the
solution can be decomposed into the orthogonal sum of a travelling wave moving with random speed and into
Gaussian fluctuations. Our results extend corresponding results obtained for stochastic neural field equations to the
present class of stochastic dynamics.