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A $\phi$-exponential distribution is a generalization of an exponentialdistribution associated to functions $\phi$ in an appropriate class, and thespace of $\phi$-exponential distributions has a dually flat structure. We studyfeatures of the space of $\phi$-exponential distributions, such as theconvexity in Wasserstein geometry and the stability under an evolutionequation. From this study, we provide the new characterizations to the space ofGaussian measures and the space of $q$-Gaussian measures.

Behaviors of $\phi$-exponential distributions in Wasserstein geometry and an evolution equation