Open AccessMarkov uniqueness of degenerate elliptic operators
Author(s) -
Derek W. Robinson,
Adam Sikora
Publication year - 2011
Publication title -
annali della scuola normale superiore di pisa. classe di scienze
Language(s) - English
Resource type - Journals
eISSN - 2036-2145
pISSN - 0391-173X
DOI - 10.2422/2036-2145.2011.3.07
Subject(s) - uniqueness , degenerate energy levels , markov chain , mathematics , pure mathematics , mathematical analysis , physics , statistics , quantum mechanics
Let Omega be an open subset of R(d) and H(Omega) = -Sigma(d)(i,j=1) partial derivative(i) c(ij) partial derivative(j) be a second-order partial differential operator on L(2)(Omega) with domain C(c)(infinity) (Omega), where the coefficients c(ij) epsilon W(l,infinity) (Omega) are real symmetric and C = (c(ij)) is a strictly positive-definite matrix over Omega. In particular, H(Omega) is locally strongly elliptic. We analyze the submarkovian extensions of H(Omega), i.e., the self-adjoint extensions that generate submarkovian semigroups. Our main result states that H(Omega) is Markov unique, i.e., it has a unique submarkovian extension, if and only if cap(Omega) (partial derivative Omega) = 0 where cap(Omega)(partial derivative Omega) is the capacity of the boundary of Omega measured with respect to H(Omega). The second main result shows that Markov uniqueness of H(Omega) is equivalent to the semigroup generated by the Friedrichs extension of H(Omega) being conservative.28 page(s