ABBM/CIR/Bessel工艺中的通用偏移和桥形
Universal Excursion and Bridge shapes in ABBM/CIR/Bessel processes
摘要：
几年前，在磁性材料的磁滞物理学的背景下，引入了一个简单的随机模型：ABBM模型。后来，ABBM模型被认为是一个广泛的不同现象的范例，被称为“噼啪作响的噪音现象”。该模型再现了这种间歇性信号的许多统计特征，如突发（或雪崩）持续时间和大小的统计，其幂律指数将证明动态是关键的。除了这样的“临界指数”，雪崩的平均形状的措施也被提出。在这里，ABBM模型的雪崩形状的平均值和波动的精确计算，表明其归一化的形状是独立于外部驱动。此外，平均值和波动的多雪崩的形状，这是一个序列的雪崩的固定的总持续时间，也计算。令人惊讶的是，这两个量（雪崩和多重雪崩归一化形状）是相同的。该结果是使用ABBM模型的精确解获得的，该ABBM模型通过精确的“时间变化”利用与Cox-Ingersoll-Ross过程（CIR）的等效性获得。介绍了这一点和其他已知的确切结果：特别是对应的ABBM/CIR模型与广义贝塞尔过程，描述动态的模的多维Ornstein-Uhlenbeck过程。因此，漂移（雪崩）和桥（多雪崩）形状分布之间的对应关系，转而适用于所有上述随机过程。简而言之：考虑到与这种扩散粒子的原点的距离，无论它是否在T之前返回，直到在时间T内返回的它的轨迹的（归一化的）平均形状（和波动）是相同的。
英文摘要：
Several years ago, in the context of the physics of hysteresis in magnetic materials, a simple stochastic model has been introduced: the ABBM model. Later, the ABBM model has been advocated as a paradigm for a broad class of diverse phenomena, baptised "crackling noise phenomena". The model reproduces many statistical features of such intermittent signals, as the statistics of burst (or avalanche) durations and sizes, with their power law exponents that would characterise the dynamics as critical. Beyond such "critical exponents", the measure of the average shape of the avalanche has also been proposed. Here, the exact calculation of average and fluctuations of the avalanche shape for the ABBM model is presented, showing that its normalised shape is independent from the external drive. Moreover, average and fluctuations of the multi-avalanche shape, that is a sequence of avalanches of fixed total duration, is also computed. Surprisingly, the two quantities (avalanche and multi-avalanche normalised shapes) are identical. This result is obtained using the exact solution of the ABBM model, obtained leveraging the equivalence with the Cox-Ingersoll-Ross process (CIR), through an exact "time change". A presentation of this and other known exact results is provided: notably the correspondence of the ABBM/CIR model with the generalised Bessel process, describing the dynamics of the modulus of the multi dimensional Ornstein-Uhlenbeck process. As a consequence, the correspondence between the excursion (avalanche) and bridge (multi-avalanche) shape distributions, turns to apply to all the aforementioned stochastic processes. In simple words: considering the distance from the origin of such diffusive particles, the (normalised) average shape (and fluctuations) of its trajectory until a return in a time T is the same, whether it has returned before T or not.
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