2021/08/07 by Raúl E. Curto, Curto, Raul E., Philipp di Dio +1 · 1 citation
Mathematics · #47 #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · doi:10.48550/arxiv.2108.03505
openalex publication_date 2021/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a new connection between the classical theory of full and truncated moment problems and the theory of partial differential equations, as follows. For the classical heat equation ∂t u = νΔu, with initial data u0 \inS(ℝn), we first compute the moments sα(t) of the unique solution u ∈ S(ℝn). These moments are polynomials in the time variable, of degree comparable to α, and with coefficients satisfying a recursive relation. This allows us to define the polynomials for any sequence, and prove that they preserve some of the features of the heat kernel. In the case of moment sequences, the polynomials trace a curve (which we call the heat curve) which remains in the moment cone for positive time, but may wander outside the moment cone for negative time. This provides a description of the boundary points of the moment cone which are also moment sequences. We also study how the determinacy of a moment sequence behaves along the heat curve. Next, we consider the transport equation ∂t u = ax ⋅ ∇ u, and conduct a similar analysis. Along the way we incorporate several illustrating examples.