Given two classes, how can I probabilistically test for equivalent behavior

Let's say I have two classes which implement the same basic API, and I want to test that they are "stochastically equivalent"¹, over at least over a subset of their methods.

E.g., I write my own "list" class foo:list and rather than painstakingly writing a bunch of unit tests for it, I want to compare it to std::list as a reference. That is, any sequence of operations on foo::list should produce the same results as the same sequence of std::list.

I'm OK listing the names of the operations, but hopefully not much more boilerplate than that. A generic solution that can be applied to other pairs of "behaviorally equivalent" classes is ideal.

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¹ By "stochastically equivalent" I mean that no differences are observed over many series of operations, which obviously falls short of a complete proof of equivalence.