Can one condition a killed random walk to survive?

We consider the simple random walk on ℤ^d killed with probability p(|x|) at site x for a function p decaying at infinity. Due to recurrence in dimension d=2, the killed random walk (KRW) dies almost surely if p is positive, while in dimension d ≥ 3 it is known that the KRW dies almost surely if and only if ∫_0^∞rp(r)dr = ∞, under mild technical assumptions on p. In this paper we consider, for any d ≥ 2, functions p for which the KRW dies almost surely and we ask ourselves if the KRW conditioned to survive is well-defined. More precisely, given an exhaustion (Λ_R)_R ∈ℕ of ℤ^d, does the KRW conditioned to leave Λ_R before dying converges in distribution towards a limit which does not depend on the exhaustion? We first prove that this conditioning is well-defined for p(r) = o(r^-2), and that it is not for p(r) = min(1, r^-α) for α∈ (14/9,2). This question is connected to branching random walks and the infinite snake. More precisely, in dimension d=4, the infinite snake is related to the KRW with p(r) ≍ (r^2log(r))^-1, therefore our results imply that the infinite snake conditioned to avoid the origin in four dimensions is well-defined.