This question is in some sense dual to the one asked in Is there an introduction to probability theory from a structuralist/categorical perspective? since contrary to the OP who asks for references about a categorical treatment of probability theory I'm interested in a probabilistic treatment of category theory. I wonder whether concepts such as random morphisms or functors have been considered so far, for example in a "dynamical graph theoretic" perspective. The idea is to build categories depending probabilistically on objects and morphisms of other classical, "deterministic" categories. My goal is to apply this kind of considerations to linguistics to understand the way a given language emerges from the way of life of its speakers and how it influences it back. I'm indeed convinced that no proper translation tool can exist without taking into account accurately the actual existence/life of the speakers of the considered languages. I'm sorry if MO is not the right place to ask such a question, but I have no idea where I could ask it otherwise as far as I feel the need for categories to tackle my problem (maybe a forum about computer science or artificial intelligence would be more appropriate, if so, feel free to tell me about such a website).
Thanks in advance for any reference and/or insight.
Edit: here comes an illustration of what I mean: