Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions

We consider the supercritical finite measure-valued diffusion, $X(t)$, whose log-Laplace equation is associated with the semilinear equation uₜ =Lu = β u - α u², where α, β> 0 and L = 1/2 ∑ i,j=1ᵈ ai,j (∂ xᵢ ∂ xⱼ)) = ∑_ i=1 ᵈ bᵢ (∂ / ∂ xᵢ). A path X() is said to survive if X(t) ≡ 0, for all t≥ 0. Since β> 0, P_μ (X() survives) $>0$, for all 0≡ μ ∈ M(Rᵈ), where M(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 meausre as follows. The support is recurrent if P _ μ (X(t,B)>0, for some t ≥ 0 | X() survives) =1, for every 0 ≡ μ ∈ M(Rᵈ) and every open set B ⊂ Rᵈ. For d≥ 2, the support is transient if P_μ(X(t,B)>0, for some t ≥ 0 |X () survives) $<1$, for every μ ∈ M(Rᵈ) and bounded B⊂ Rᵈ which satisfy (μ) B̄ = ∅. A similar definition taking into account the topology of R¹ is given for $d=1$. The support exhibits local extinction if for each μ ∈ M(Rᵈ) and each bounded B⊂ Rᵈ, there exists a P_μ-almost surely finite random time ζB such that $X(t,B) = 0$, for all t≥ ζB. Criteria for transience, recurrence and local extinction are developed in this paper. Also studied is the asymptotic behavior as t → ∞ of E_μ ∫₀ᵗ ψ, X(s) ds, and of E_μ g,X(t), for 0≤ g, ψ ∈ Cc(Rᵈ), where f, X(t) ≡ ∫Rᵈ f(x) X(t,dx). A number of examples are given to illustrate the general theory.