2025/09/29 by Zachhuber, Immanuel
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)
paper · doi:10.48550/arxiv.2509.24950
In this note we show how one can use recently gained insights from the study of singular SPDEs, more particularly the study of singular operators via the theory of Paracontrolled Distributions, to construct domains for (singular) elliptic operators. Formally we consider A (u) "='' (1 - Δ) u + ∇ V ⋅ ∇ u + ξu + div (ρu), where V ∈ Cδ, ξ∈ C- 2 + δ, ρ∈ C- 1 + δ,divρ= 0 and which satisfy a structural assumption that is notably satisfied when ξ is a "sub-critical noise". We also show that under this assumption, one can construct a continuous change of variables Θ which satisfies A Θ- (1 - Δ) ∈ L (H2 ; Hδ') which allows us to define A rigorously and parametrise a domain. Moreover, for suitably regularised operators Aε (u) := (1 - Δ) u + ∇ Vε ⋅ ∇ u + (ξε + cε) u + div (ρε ⋅ u), we show that for a strongly converging regularised change of variables Θε → Θ we have Aε Θε → A Θ in L (H2 ; L2) which in particular implies norm resolvent convergence to a limiting closed operator.