Zusammenfassung für den Vortrag am 31.01.2018 (11:15 Uhr)

Arbeitsgemeinschaft ANGEWANDTE ANALYSIS

Pavlos Tsatsoulis (University of Warwick)
Exponential loss of memory for the dynamic Φ₂⁴ with small noise

We consider the dynamic Φ₂⁴ model (or stochastic Allen-Cahn equation) formally given by the SPDE

where \u03be is a space-time white noise. When ξ is small the solutions of the equation spend long time intervals in metastable states before reaching equilibrium. This phenomenon is known as metastability. We discuss a coupling argument for solutions started from suitable initial conditions in space dimension 2 as a consequence of metastability. Such a result is already known in space dimension 1. The basic obstacle in our case is that classical solution theory for SPDEs is not applicable here since the non-linear term is ill-defined due to the irregularity of ξ. Hence a renormalization is required to compensate the divergences of the non-linear term. This destroys the "nice" structure of the non-linear term and a deeper analysis on the level of the so-called "remainder" term is required. An interesting application of such a coupling argument appears in the proof of the Eyring-Kramers law in the theory of metastability.