We study a generic model of the war of attrition with symmetric information and stochastic payoffs that depend on a homogeneous linear diffusion. We first show that a player's mixed Markov strategy can be represented by an intensity measure over the state space together with a subset of the state space over which the player concedes with probability 1. We then show that, if players are asymmetric, then, in all mixed-strategy Markov-perfect equilibria, these intensity measures must be discrete, and characterize any such equilibrium through a variational system for the players' value functions. We illustrate these findings by revisiting the standard model of exit in a duopoly under uncertainty and construct a mixed-strategy Markov-perfect equilibrium in which attrition takes place on path despite firms having different liquidation values. We show that firms' stock prices comove negatively over the attrition zone and exhibit resistance and support patterns documented by technical analysis.