Time-Dependent Moments From the Heat Equation and a Transport Equation

Abstract

AbstractWe 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 $\partial _t u = {\nu } \Delta u$, with initial data $u_0 \in {\mathcal {S}}(\mathds {R}^n)$, we first compute the moments $s_{\alpha }(t)$ of the unique solution $u \in {\mathcal {S}}(\mathds {R}^n)$. These moments are polynomials in the time variable, of degree comparable to $\alpha $, 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 $\partial _t u = ax \cdot \nabla u$ and conduct a similar analysis. Along the way we incorporate several illustrating examples. We show that while $\partial _t u = {\nu }\Delta u + ax\cdot \nabla u$ has no explicit solution, the time-dependent moments can be explicitly calculated.