Suppose L is a second-order elliptic differential operator in R d and D is a bounded, smooth domain in R d . Let 1 < α ≤ 2 and let Γ be a closed subset of ∂D. It is known [13] that the following three properties are equivalent:
(α) Γ is ∂-polar; that is, Γ is not hit by the range of the corresponding (L, α)-superdiffusion in D;
(β) the Poisson capacity of Γ is equal to 0; that is, the integral
α is equal to 0 or ∞ for every measure ν, where ρ(x) is the distance to the boundary and k(x, y) is the corresponding Poisson kernel; and
(γ) Γ is a removable boundary singularity for the equation
We investigate a similar problem for a parabolic operator in a smooth cylinder Q = R+×D. Let Γ be a compact set on the lateral boundary of Q. We show that the following three properties are equivalent:
(a) Γ is G-polar; that is, Γ is not hit by the graph of the corresponding (L, α)-superdiffusion in Q;
(b) the Poisson capacity of Γ is equal to 0; that is, the integral Q ρ(x)dr dx Γ k(r, x; t, y)ν(dt, dy) α is equal to 0 or ∞ for every measure ν, where k(r, x; t, y) is the corresponding (parabolic) Poisson kernel; and
(c) Γ is a removable lateral singularity for the equation u + Lu = u α in Q; that is, if u ≥ 0 and u + Lu = u α in Q and if u = 0 on ∂Q \ Γ and on {∞} × D, then u = 0.