We study a generic family of two-player continuous-time nonzero-sum stopping games modeling a war of attrition under complete information and stochastic payoffs depending on a homogeneous linear diffusion. We first show that any Markovian mixed strategy can be represented by a pair $(\mu,S)$ where $\mu$ is a measure on the state space representing the stopping intensity and $S$ is a subset of the state space in which the player stops with probability 1. We prove that if the players are asymmetric, then in all the Markov Perfect equilibria in mixed strategies, the measures $\mu$ have to be essentially discrete, and we characterize any such equilibrium through a variational system satisfied by the equilibrium payoffs of the players. This contrasts with the existing literature in which only pure equilibria or mixed equilibria using absolutely continuous stopping intensities in symmetric models were studied. We illustrate this result by revisiting the model of exit in a duopoly under uncertainty, and exhibit an equilibrium in mixed strategies with asymmetric players in which the players enter a war of attrition and delay their exit decision compare to the pure equilibria, as it happens in the mixed equilibria in classical models with deterministic payoffs. (Work with Jean-Paul Décamps and Thomas Mariotti)