Scarpa, Luca; (2018) A variational approach to some classes of singular stochastic PDEs. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

This thesis contains an analysis of certain classes of parabolic stochastic partial differential equations with singular drift and multiplicative Wiener noise. Equations of this type have been studied so far only under rather restrictive hypotheses on the growth and smoothness of the drift. By contrast, we give here a self-contained treatment for such equations under minimal assumptions. The first part of the thesis is focused on semilinear SPDEs with singular drift. In particular, the nonlinearity in the drift is the superposition operator associated to a maximal monotone graph everywhere defined on the real line, on which neither continuity nor growth assumptions are imposed. The hypotheses on the diffusion coefficient are also very general. First of all, well-posedness is established for the equation through a combination of variational techniques and a priori estimates. Secondly, several refined well-posedness results are provided, allowing the initial datum to be only measurable and the diffusion coefficient to be locally Lipschitz-continuous. Moreover, existence, uniqueness and integrability properties of invariant measures for the Markovian semigroup generated by the solution are proved. Furthermore, the associ- ated Kolmogorov equation is studied in Lp spaces with respect to the invariant measure and the infinitesimal generator of the transition semigroup is characterized as the closure of the corresponding Kolmogorov operator. The second part of the thesis focuses on equations with monotone singular drift in divergence form. Due to rather general assumptions on the growth of the nonlinearity in the drift, which, in particular, is allowed to grow faster than polynomially, existing techniques are not applicable. Equations of this type are typically doubly nonlinear, making their treatment more challenging in comparison to the semilinear case. Well-posedness for such equations is established in several cases, suitably generalizing the techniques for semilinear equations to an abstract generalized variational setting.

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