I am looking for a popular stochastic model employed for a trajectory of a fish which tries to keep staying at the initial position against water pressure from time-varying directions.

The trivial one is a random walk model with behavior of going back to the initial position.

$ x_{t+1} = x_t + \frac{x_0 - x_t}{|x_0 - x_t|}v_t + \xi_t $

$v_t = \begin{cases}v_{max} & \text{ if } |x_0 - x_t| > v_{max} \\ |x_0 - x_t| & \text{else}\end{cases}$

Here, $x_t \in \mathbb{R}^3$ is the position of the fish at time $t$, $v_t$ means a speed of the fish during returning to $x_0$, and $\xi_t$ is random noise that represents the water pressure from the various direction.

I checked several textbooks of stochastic process and Wikipedia pages to find a model suitable for this kind of behavior. However, I could not find the good one. To the best of my knowledge, "correlated random walk" treats the similar situation of mine. However, I feel that model is too rich for modeling the behavior of fish staying at the same position.

Please let me know if you know the popular stochastic model suitable for modeling this kind of staying behavior?


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