du Buisson Johan, Mnyulwa Thamu D P, Touchette Hugo
Institute of Theoretical Physics, Department of Physics, Stellenbosch University, Stellenbosch 7600, South Africa.
Department of Mathematical Sciences, Stellenbosch University, Stellenbosch 7600, South Africa.
Phys Rev E. 2023 Oct;108(4-1):044136. doi: 10.1103/PhysRevE.108.044136.
The area enclosed by the two-dimensional Brownian motion in the plane was studied by Lévy, who found the characteristic function and probability density of this random variable. For other planar processes, in particular ergodic diffusions described by linear stochastic differential equations (SDEs), only the expected value of the stochastic area is known. Here we calculate the generating function of the stochastic area for linear SDEs, which can be related to the integral of the angular momentum, and extract from the result the large deviation functions characterizing the dominant part of its probability density in the long-time limit, as well as the effective SDE describing how large deviations arise in that limit. In addition, we obtain the asymptotic mean of the stochastic area, which is known to be related to the probability current, and the asymptotic variance, which is important for determining from observed trajectories whether or not a diffusion is reversible. Examples of reversible and irreversible linear SDEs are studied to illustrate our results.
平面中二维布朗运动所围成的区域由列维进行了研究，他求出了这个随机变量的特征函数和概率密度。对于其他平面过程，特别是由线性随机微分方程（SDEs）描述的遍历扩散，仅知道随机面积的期望值。在此，我们计算线性随机微分方程的随机面积的生成函数，它可与角动量的积分相关联，并从结果中提取在长时间极限下表征其概率密度主导部分的大偏差函数，以及描述在该极限下大偏差如何产生的有效随机微分方程。此外，我们得到了已知与概率流相关的随机面积的渐近均值，以及对于从观测轨迹确定扩散是否可逆很重要的渐近方差。研究了可逆和不可逆线性随机微分方程的例子以说明我们的结果。
