whose log-Laplace equation is associated with the semilinear equation Ut = Lu + flu - au2, where a,,8 > 0, and L = 1 Ed j= l aij (d92/( xXi dxj)) + Ed= 1 bi (d/8xi). A path X() is said to survive if X(t) # 0, for all t 2 0. Since / > 0, P],(X(Q) survives) > 0, for all 0 W ,u e4(Rd), where (Rd) denotes the space of finite measures on Rd. We define transience, recurrence and local extinction for the support of the supercritical measure-valued diffusion starting from a finite measure as follows. The support is recurrent if P,,(X(t, B) > 0, for some t 2 0 I X() survives) = 1, for every 0 # ,u eJ(Rd) and every open set B c Rd. For d ? 2, the support is transient if P,(X(t, B) > 0, for some t 2 0 I X() survives) < 1, for every /ut EJ(Rd) and bounded B c Rd which satisfy supp( /u n B = 0. A similar definition taking into account the topology of R1 is given for d = 1. The support exhibits local extinction if for each ,t e.4(Rd) and each bounded B c Rd, there exists a P,L-almost surely finite random time ;B such that X(t, B) = 0, for all t 2 ;B. Criteria for transience, recurrence and local extinction are developed in this paper. Also studied is the