First of all, I want to state that I'm not an expert in the game theory and searching the references for the game I just made up. Solving this game by itself seems like a decent project for strong highschool / undergrad students and I would like to suggest it to some people, but I'm not sure whether anything is known about it already and need to familiarize with the topic.

The game is a zero-sum probabilistic game with two players: runner and catcher.

Runner moves in some space, say a line or a plane with limited speed $v$ and arbitrary direction.

Catcher knows the exact position of a runner. Once catcher can declare some circle of radius $R$, and after the time $t$ passes, if the runner is in the circle, the catcher wins. Otherwise, runner wins.

Runner doesn't know the declaration moment.

We also state that if for $T >> t$ catcher does not declare anything, runner wins automatically (but I think that runner's behavior should actually have some limit for $T \rightarrow \infty$).

I've tried to tackle the problem for the following discrete model: the time is discrete, runner moves 1 step left or right each turn, catcher declares 1 possible position, $t = 2$ turns. Then, there is a strategy for a runner which allows to escape with probability $\frac{\sqrt{5} - 1}{2} \approx 0.62$ independent on a catcher behavior. I believe it is optimal but didn't prove it fully. Note that escape with probability $> \frac{2}{3} \approx 0.66$ is not possible due to the following catcher strategy: choose $3$ possible positions of a runner on a second turn with equal probability.

Note also that the symmetric random walk allows the escape with probability $\frac{1}{2}$ which is much worse.

Does anybody know something about this game or any possible references?