Absolute continuity results for superdiffusions with applications to differential equations
Eugene B. Dynkin · Comptes Rendus Mathématique · 2004
Let X =( X D , P μ ) be a superdiffusion in a domain E ⊂ ℝ d . We introduce a germ σ -algebra ℱ E - at the boundary of E and we prove that, on this σ -algebra, P μ 1 is absolutely continuous with respect to P μ 2 if μ 1 and μ 2 are concentrated on compact subsets of E . In combination with previous results of Dynkin, Kuznetsov and Mselati, this leads to a complete classification of positive solutions of equation Δ u = u α in a bounded domain E of class C 4 for the case 1< α ⩽2.