Singular SPDEs

Variational methods for singular SPDEs

A common problem in the theory of singular Stochastic Partial Differential Equations (SPDEs) is that the driving noise of these equations is very irregular and their non-linear terms are not classically defined.

We are interested in the study of SPDEs that arise as the Euler-Lagrange equation of variational problems for energy functionals of the type $$ \mathcal{E}(u) = \mathcal{Q}_{L}(u) + \mathcal{G}(u) - \xi(u) $$ where \(\mathcal{Q}_L\) is the quadratic form associated to some linear operator \(L\), \(\mathcal{G}\) some higher order nonlinearity, and \(\xi\) is Gaussian white noise (acting on \(L^2(\mathbb{R}^d)\)). Stationary points of \(\mathcal{E}\) satisfy the equation $$ L u + D\mathcal{G}(u) = \xi \quad \text{in a weak sense.} $$ Typically, even though the operator \(L^{-1}\) is regularising, solutions are not expected to be regular enough to give sense to the nonlinear expressions in \(D\mathcal{G}(u)\). These have to be constructed by "off-line" stochastic arguments (e.g. through Wick renormalisation as in Da Prato-Debussche or via Hairer's regularity structures ). Once these objects have been given sense in the right spaces, the problem can be amenable to a treatment via fixed-point arguments, yielding short-time or small-data existence.

In contrast, when looking for a long-time or large-data existence theory, the variational structure of the equation can provide access to a construction of solutions via minimisation of the energy \(\mathcal{E}\). However, it turns out that - generically - the energy of the system is on average \(-\infty\), so that the energy functional has to be renormalised in a suitable way.

A natural approach to the renormalisation of the energy is by replacing the singular noise term \(\xi\) by suitable approximations \(\xi_{\ell}\) that as \(\ell \downarrow 0\) converge to \(\xi\) in an appropriate sense, for instance almost surely or in law. The energy corresponding to the regularisation of the noise diverges as the regularisation is removed, but one can isolate the divergent term, so that the resulting renormalised energy is amenable to \(\Gamma\)-convergence techniques.

The Magnetisation Ripple

In [1] we study the so-called magnetisation ripple, which is an ubiquitous microstructure formed by the magnetisation in a thin-film ferromagnet triggered by the random orientation of the grains in the polycrystalline material. The model used to describe the magnetisation ripple was derived by Ignat-Otto within the Landau-Lifshitz theory of micromagnetics and is of the above variational type, leading to a nonlocal and nonlinear elliptic SPDE driven by white noise in two dimensions, which is singular.

To justify the universal character of the ripple, the renormalised energy functional is recovered as the \(\Gamma\)-limit (in law) from a large class of non-Gaussian approximations to white noise satisfying a so-called spectral gap inequality. A surprising feature of this approach is that successive application of the spectral gap inequality automatically leads to the right renormalisation constants that are needed to construct the singular products making up the nonlinearities in the model.

The ripple problem also has an interesting modelling perspective related to stochastic geometry: the grains can be thought of as the Voronoi cells of a random tessellation of the plane, with a random orientation attached to each cell. One can show that in a suitable rescaling, this model converges to white noise, however it does not satisfy the spectral gap inequality, rather a multiscale version of a spectral gap inequality, which reflects the fact that correlations in a Poisson-Voronoi mosaic are not finite-range, but still decay sufficiently fast.

Related publications

  1. Variational methods for an elliptic singular SPDE describing the magnetization ripple. In Preparation
    (with R. Ignat, F. Otto, P. Tsatsoulis)