We consider Brownian motion under resetting in higher dimensions for the case when the return of the particle to the origin occurs at a constant speed. We investigate the behavior of the probability density function (PDF) and of the mean-squared displacement (MSD) in this process. We study two different resetting protocols: exponentially distributed time intervals between the resetting events (Poissonian resetting) and resetting at fixed time intervals (deterministic resetting). We moreover discuss a general problem of the invariance of the PDF with respect to the return speed, as observed in the one-dimensional system for Poissonian resetting, and show that this one-dimensional situation is the only one in which such an invariance can be found. However, the invariance of the MSD can still be observed in higher dimensions.

中文翻译：

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高维非瞬时复位下的布朗运动

当粒子以恒定速度返回原点时，我们将在更大的维数下考虑布朗运动。我们调查在此过程中概率密度函数（PDF）和均方位移（MSD）的行为。我们研究了两种不同的重置协议：重置事件之间的指数分布时间间隔（泊松重置）和以固定时间间隔重置（确定性重置）。此外，我们讨论了在Poissonian重置的一维系统中观察到的PDF相对于返回速度的不变性的一般问题，并表明这种一维情况是唯一可以实现这种不变性的情况。找到了。但是，仍然可以在更高维度上观察到MSD的不变性。