By Christophette Blanchet-Scalliet
By Philip Protter
By Vlad Bally
By Felix Otto
Appears in collection : Advances in Nonlinear Analysis and Nonlinear Waves, a conference in honor of Frank Merle
Singular stochastic PDE are those stochastic PDE in which the noise is so rough that the nonlinearity requires a renormalization. The guiding principle of renormalization is to preserve as many symmetries of the solution manifold as possible. We follow the typical approach of mathematical physics, and of Hairer’s regularity structures, which provides a formal series expansion of a general solution. However, we advocate a more geometric/analytic than combinatorial version of this approach: Instead of appealing to an expansion in- indexed by trees, we consider all partial derivatives w. r. t. the “constitutive” function defining the nonlinearity. Instead of a Gaussian calculus guided by Feynman diagrams arising from pairing nodes of two trees, we consider derivatives w. r. t. the noise, i.e. Malliavin derivatives. This calculus allows to characterization the expansion without divergent terms; in conjunction with the spectral gap estimate, it provides a natural path toward stochastic estimates. This is joint work with P. Linares, M. Tempelmayr, and P. Tsatsoulis, based on work with J. Sauer, S. Smith, and H. Weber.