In this paper, we construct a probabilistic model of a sliding mode. This model is based on the moment a random walk with positive jumps crosses a certain critical level. It is assumed that the jump magnitude has a geometric distribution. If the initial state is negative and the critical level is zero, then after crossing this level, a random walk begins in the opposite direction until it crosses zero again. As a result, motion orthogonal to the slip line is defined as a regenerative process, in which the moments of regeneration are the moments of zero crossings from right to left. An estimate of the Qi Fan metric of the maximum deviation of this random walk over a certain time interval is constructed under the assumption that the time and magnitude of the jumps are reduced by a factor of m. This estimate is found to be of the order of lnm/m as m→∞ and characterizes the deviation of a random trajectory orthogonal to the slip line. In the model of motion along a slip line, its velocity is assumed to have fixed values when the trajectory of motion orthogonal to the slip line is above or below zero. Using the central limit theorem for the integral of a regenerative process, an estimate of the non-uniformity of motion of a random trajectory along the slip line is constructed. It is found that the characteristic magnitude of this non-uniformity is of the order of 1/m as m→∞. This indicates that the accumulation of random errors during motion along the slip line is significantly faster than during motion orthogonal to the slip line.
Gurami Tsitsiashvili (Mon,) studied this question.