Let be an open subset of and be a second-order partial differential operator on with domain , where the coefficients are real symmetric and is a strictly positive-definite matrix over . In particular, is locally strongly elliptic. We analyze the submarkovian extensions of , i.e., the self-adjoint extensions that generate submarkovian semigroups. Our main result states that is Markov unique, i.e., it has a unique submarkovian extension, if and only if where is the capacity of the boundary of measured with respect to . The second main result shows that Markov uniqueness of is equivalent to the semigroup generated by the Friedrichs extension of being conservative.
@article{ASNSP_2011_5_10_3_683_0, author = {Robinson, Derek W. and Sikora, Adam}, title = {Markov uniqueness of degenerate elliptic operators}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {683--710}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 10}, number = {3}, year = {2011}, mrnumber = {2905383}, zbl = {1259.47026}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2011_5_10_3_683_0/} }
TY - JOUR AU - Robinson, Derek W. AU - Sikora, Adam TI - Markov uniqueness of degenerate elliptic operators JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2011 SP - 683 EP - 710 VL - 10 IS - 3 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2011_5_10_3_683_0/ LA - en ID - ASNSP_2011_5_10_3_683_0 ER -
%0 Journal Article %A Robinson, Derek W. %A Sikora, Adam %T Markov uniqueness of degenerate elliptic operators %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2011 %P 683-710 %V 10 %N 3 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2011_5_10_3_683_0/ %G en %F ASNSP_2011_5_10_3_683_0
Robinson, Derek W.; Sikora, Adam. Markov uniqueness of degenerate elliptic operators. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 3, pp. 683-710. http://www.numdam.org/item/ASNSP_2011_5_10_3_683_0/
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