Integral Printing

Transition probabilities for degenerate diffusions arising in population genetics

“…The proof that we provide is a small modification of an argument communicated to us by M.V. Safonov, [24].…”

“…The first result is a Landis-type growth lemma which was proved by M.V. Safonov, [24], in the two-dimensional case (that is, we take n = 2 and m = 0 in the statement of Lemma A.1). Given r, µ > 0 and a bounded function u, we introduce the notation: We consider a differential operator A defined on B ∞ R ⊂ S n,m by…”

“…We are able to apply Theorem 1.9 to give the proof of Theorem 1.10, when we use in addition a Landis-type growth lemma proved by M.V. Safonov, [24], and a regularity result concerning the hitting probabilities of tangent boundary components. We prove the last two results that we mentioned in Appendix A.…”

“…Lemma A.1 (Growth lemma). [24] Let n, m be non-negative integers such that n ≥ 2 and let 0 < R, ν < 1. Suppose that the coefficients of the operator A in (A.3) are bounded continuous functions onB ∞ R and that there are positive constants, δ and K, such that for all 1 ≤ i, j ≤ n and 1 ≤ l, k ≤ m, we have that…”

“…Lemma A.3 (Growth lemma with n = 2 and m ≥ 0). [24] In the statement of Lemma A.1 assume that n = 2, m ≥ 0, i = 1, and j = 2. Then the conclusion of Lemma A.1 holds.…”