In mechanics, and in particular in shape analysis, taking into account the underlying geometric properties of a problem to model it is often crucial to understand and solve it. This approach has mostly been applied for isolated systems, or for systems interacting with a well-defined, deterministic environment. In this talk, I want to discuss how to go beyond this deterministic description of isolated systems to include random interactions with an environment, while retaining as much as possible the geometric properties of the isolated systems. I will discuss examples from geometric mechanics to shape analysis, ranging from interacting rigid bodies with a heath bath to uncertainties quantification in computational anatomy.