A zero–one law of almost sure local extinction for -super-Brownian motion

This paper considers the following generalized almost sure local extinction for the d -dimensional ( 1 + β ) -super-Brownian motion X starting from Lebesgue measure on R d . For any t ≥ 0 write B g ( t ) for a closed ball in R d with center at 0 and radius g ( t ) , where g is a nonnegative, nondecreasing and right continuous function on [ 0 , ∞ ) . Let τ ≔ sup { t ≥ 0 : X t ( B g ( t ) ) > 0 } . For d β < 2 , it is shown that P { τ = ∞ } is equal to either 0 or 1 depending on whether the value of the integral ∫ 1 ∞ g ( y ) d y − 1 − 1 / β d y is finite or infinite, respectively. An asymptotic upper bound for P { τ > t } is found when P { τ < ∞ } = 1 .