Suppose, we have $k$ urns each starting with $n$ balls (all balls are of the same color). At each step, we draw balls from each urn, where each ball could be drawn independently at random with probability $p$. After we picked those balls. They are returned to the urns, with each ball returned independently at random to one of the urns.
I'm interested in some directions to existing models, that describe the long-term behavior of such a system.
It resembles a random-walk on $n-$regular simplex, but the increments are also random and I don't see a way to model it as picking one ball at a time.
Let $X_i(t)$ - be a number of balls in urn $i$ at discrete time $t$, then among others $$\lim\limits_{t \to \infty} \mathbb{P}\Big( \min X_i(t) \geq \frac{k-1}{k} n \Big) \geq ~?$$ is of interest.
Since, the model have a nice symmetries regarding the urns, I would expect the answer to be $1$, probably for any number $\alpha < 1$, where above $\alpha = \frac{k-1}{k}$. Is this true?