Regularisation by multiplicative noise for reaction-diffusion equations with distributional drift

In this thesis we deal with regularisation by noise phenomena in partial differential equations caused by multiplicative noises. In particular, we study the solvability of stochastic reaction-diffusion equations in the case when the “drift” or “reaction term” is so irregular that it is not even a function, but merely a distribution in the Schwartz-sense. We provide the definition of solution, which is a priori not well-understood. We use Malliavin calculus to obtain estimates related to the density of the solution of the driftless equation, such as quantitative bounds on all Malliavin derivatives of the solution, and a quantitative nondegeneracy result for the first Malliavin derivative. We use these results to prove an “integration by parts” result for computing expectations of functions of the driftless equation. We also show the Lipschitz continuity of the Malliavin derivatives of the driftless equation in the initial condition, and quantify how well the driftless equation approximates the general case with (possibly distributional) drift. We combine the aforementioned results with state-of-the-art stochastic sewing techniques to prove a well-posedness result for the stochastic reaction-diffusion equation with distributional drift, derive stability estimates, and establish the temporal and spatial regularity of the solution. Moreover, we provide bounds on the H\"older norms of the density of solution of the driftless equation, and all of its derivatives.