A priori bounds for fractional SPDEs

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The principal aim of Salvador'r research project is to prove certain a priori estimates for a dynamic fractional Phi4 model. This amounts to studying a so-called "singular SPDE", i.e. a partial differential equations with an extremely irregular noise term on the right hand side. The analysis of these equations requires a delicate mix of probabilistic and analytic technique, that has only been developed in the last years (e.g. Hairer's theory of Regularity Structures). The Phi4 model is a well known model from Mathematical Physics with relevance both in statistical mechanics and in constructive field theory. In this classical model, a priori bounds have been derived in the last years. The aim of the research project is to extend these results to a "fractional equation", i.e. the Laplacian in the heat operator is replaced by a suitable fractional power. In the mathematical Physics literature this modification had been proposed and studied by Brydges, Mitter and Scoppola. The model is interesting, because it gives a natural way to define Phi4 in fractional dimensions and thereby allows to approach the critical threshold of four dimensions. The problem comes in different difficulties depending on the choice of fractional power (the lower, the more difficult) and spatial domain (infinite domains are harder than finite domains). As a first step, Salvador will establish the correct estimates in a case that - in terms of scaling - behaves similar to the classical theory on a two-dimensional torus. The extension to more irregular situations and to infinite domains will come after.