Well-posedness of stochastic evolution equations with H\"older continuous noise

We show existence and pathwise uniqueness of probabilistically strong solutions to a pseudomonotone stochastic evolution problem on a bounded domain Dᵈ, d, with homogeneous Dirichlet boundary conditions and random initial data u₀∈ L²(Ω;L²(D)). The main novelty is the presence of a merely H\"older continuous multiplicative noise term. In order to show the well-posedness, we simultaneously regularize the H\"older noise term by inf-convolution and add a perturbation by a higher order operator to the equation. Using a stochastic compactness argument we may pass to the limit and we obtain first a martingale solution. Then by a pathwise uniqueness argument we get existence of a probabilistically strong solution.