Let L be the hypoelliptic Ornstein–Uhlenbeck operator associated with the pair of matrices (A,B). In 2004, Priola and Zabczyk proved the following Liouville-type theorem: every bounded entire solution of Lu = 0 is constant if and only if (∗) every eigenvalue of B has real part less than or equal to zero. This remarkable result raised the following problem, which is still not completely solved: if condition (∗) holds, is it true that every non-negative entire solution of Lu = 0 is constant? In this note, along with a review of the current state of research on this problem, we present some recent new results.
ON A LONG STANDING CONJECTURE: POSITIVE LIOUVILLE THEOREM FOR HYPOELLIPTIC ORNSTEIN–UHLENBECK OPERATORS
Tralli G.
2025
Abstract
Let L be the hypoelliptic Ornstein–Uhlenbeck operator associated with the pair of matrices (A,B). In 2004, Priola and Zabczyk proved the following Liouville-type theorem: every bounded entire solution of Lu = 0 is constant if and only if (∗) every eigenvalue of B has real part less than or equal to zero. This remarkable result raised the following problem, which is still not completely solved: if condition (∗) holds, is it true that every non-negative entire solution of Lu = 0 is constant? In this note, along with a review of the current state of research on this problem, we present some recent new results.File in questo prodotto:
Non ci sono file associati a questo prodotto.
I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
