2018/07/20 by Luciana Angiuli, Angiuli, Luciana, Simone Ferrari +3
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · doi:10.48550/arxiv.1807.07780
openalex publication_date 2018/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a separable Hilbert space endowed with a non-degenerate centred Gaussian measure γ and let λ1 be the maximum eigenvalue of the covariance operator associated with γ. The associated Cameron--Martin space is denoted by H. For a sufficiently regular convex function U:X→ℝ and a convex set Ω⊆ X, we set ν:=e-Uγ and we consider the semigroup (TΩ(t))t≥ 0 generated by the self-adjoint operator defined via the quadratic form (φ,ψ)↦ ∫Ω⟨ DHφ,DHψ⟩Hdν, where φ,ψ belong to D1,2(Ω,ν), the Sobolev space defined as the domain of the closure in L2(Ω,ν) of DH, the gradient operator along the directions of H. A suitable approximation procedure allows us to prove some pointwise gradient estimates for (TΩ(t))t≥ 0. In particular, we show that |DH TΩ(t)f|Hp≤ e^- p λ1-1 t(TΩ(t)|DH f|pH), tgt;0, ν\textrm -a.e. in Ω, for any p∈ [1,+∞) and f∈ D1,p(Ω,ν). We deduce some relevant consequences of the previous estimate, such as the logarithmic Sobolev inequality and the Poincaré inequality in Ω for the measure ν and some improving summability properties for (TΩ(t))t≥ 0. In addition we prove that if f belongs to Lp(Ω,ν) for some p∈(1,∞), then |DH TΩ(t)f|pH ≤ Kp t-(p)/(2) TΩ(t)|f|p, tgt;0, ν-a.e. in Ω, where Kp is a positive constant depending only on p. Finally we investigate on the asymptotic behaviour of the semigroup (TΩ(t))t≥ 0 as t goes to infinity.