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Let's suppose we have an active Brownian particle whose overdamped equation of motion is given by \begin{equation} \begin{cases} \dot{x} = f + \sqrt{2D}\xi_x \\ \dot{y} = \sqrt{2D}\xi_y \end{cases} \end{equation} where $f$ is a positive constant and $\xi_i$ is a Gaussian white noise delta-correlated in time with unit variance. The particle starts at $(x_0,y_0)$ and has to reach a target located at $(x_1,y_0)$, with $x_1>x_0$. What is the First Passage Time distribution for this process?

I guess my question is kind of related to this one:

Density of first hitting time of Brownian motion with drift

where they find that the solution is an Inverse Gaussian distribution. Is it possible to generalize this result to the case I presented? If so, how?

lorenzop
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1 Answers1

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It is well-known that points (and countable unions of points) are polar for Brownian motion in dimension greater than one. For example see this paper. The first time that your BM starting at $(x_0,x_0)$ will reach $(x_1,y_0)$ will therefore be infinite. The fact that a constant drift $f$ is added won't change the matters.

Kurt G.
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