  We consider evolutionary dynamics for population games in which players have
a continuum of strategies at their disposal. Models in this setting amount to
infinite-dimensional differential equations evolving on the manifold of
probability measures. We generalize dissipativity theory for evolutionary games
from finite to infinite strategy sets that are compact metric spaces, and
derive sufficient conditions for the stability of Nash equilibria under the
infinite-dimensional dynamics. The resulting analysis is applicable to a broad
class of evolutionary games, and is modular in the sense that the pertinent
conditions on the dynamics and the game's payoff structure can be verified
independently. By specializing our theory to the class of monotone games, we
recover as special cases existing stability results for the Brown-von
Neumann-Nash and impartial pairwise comparison dynamics. We also extend our
theory to models with dynamic payoffs, further broadening the applicability of
our framework. We illustrate our theory using a variety of case studies,
including a novel, continuous variant of the war of attrition game.
