We consider measure-valued processes Xt that solve the following martingale problem: For a given initial measure X0, and for all smooth, compactly supported test functions (Ψ),Xt(φ) =X0((Ψ)) +(1/2) Xs(△φ)ds +θ∫ Xs(φ)ds-∫to Xs(Lsφ)ds + Mt(φ).Here Ls(x) is the local time density process associated with X.and Mt(φ) is a martingale with quadratic variation [M(φ)]t =∫to Xs(φ2) ds.We show that there exist critical values θc(d) ∈ (0,∞) for d =2,3 such that if θ > θc(d), then the solution suwives forever with positive probability, but if θ < θc(d), then the solution dies out in finite time with probability 1.For d =1 we prove that the solution dies out almost surely for all values of θ.Ve also show that in dimensions d =2, 3 the process dies out locally almost surely for any value of θ, that is, for any compact set K, the process Xt(K) =0 eventually.