The Backward Equation

Abstract

The backward solution u(p, t) expresses the probability of having started in some p ∈â€‰Δ n at the negative time t conditional upon being in a certain state u(p, 0) = f(p) at time t = 0, i.e. having reached the corresponding (generalised) target set . It becomes a parabolic equation upon time reversal, that is, replacing t by − t. We can then treat u(p, 0) = f(p) as the initial condition at time t = 0. In view of the biological model behind the Kolmogorov backward equation , however, we shall work with negative time and call u(p, 0) = f(p) a final condition.