Abstract
This note contains a representation formula for positive solutions of linear degenerate second-order equations of the form $$ \partial_t u (x,t) = \sum_{j=1}^m X_j^2 u(x,t) + X_0 u(x,t) \qquad (x,t) \in \mathbb{R}^N \times\, ]- \infty ,T[,$$ proved by a functional analytic approach based on Choquet theory. As a consequence, we obtain Liouville-type theorems and uniqueness results for the positive Cauchy problem.
Highlights
In this article we consider second-order partial differential operators of the form mL u := ∂tu − Xj2u − X0u j=1 in RN+1. (1.1)Points z ∈ RN+1 are denoted by z = (x, t), where x ∈ RN, t ∈ R
We first derive from (H*) a Harnack inequality for the operator L − λ, where L is of the form (1.1) and λ is a real constant
We apply the well-known argument to show that the uniqueness for the positive Cauchy problem is equivalent to the uniqueness of the positive Cauchy problem with the zero initial condition
Summary
In this article we consider second-order partial differential operators of the form m. We note that the restricted uniform Harnack inequality (Proposition 1.2) is used in the proof of our separation principle to construct Harnack chains along the path γ(s) = exp (s (ω · X + Y )) (x0, t0). We have the following result, useful in the study of stratified Lie groups and the Mumford operator It is weaker than Theorem 1.4 in that the right-invariance of solutions is not assumed to hold for every positive s.
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