The Hawkes self-excited point process provides an efficient representation of the bursty intermittent dynamics of many physical, biological, geological, and economic systems. By expressing the probability for the next event per unit time (called “intensity”), say of an earthquake, as a sum over all past events of (possibly) long-memory kernels, the Hawkes model is non-Markovian. By mapping the Hawkes model onto stochastic partial differential equations that are Markovian, we develop a field theoretical approach in terms of probability density functionals. Solving the steady-state equations, we predict a power law scaling of the probability density function of the intensities close to the critical point $n=1$ of the Hawkes process, with a nonuniversal exponent, function of the background intensity ${\nu }_0$ of the Hawkes intensity, the average timescale of the memory kernel and the branching ratio $n$. Our theoretical predictions are confirmed by numerical simulations.

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自激霍克斯过程强度的非通用幂律分布：一种场论方法

霍克斯自激点过程可以有效地表示许多物理，生物，地质和经济系统的间歇性间歇动力学。通过将单位时间内发生下一个事件的概率（称为“强度”）表示为（可能是）长内存内核的所有过去事件的总和，霍克斯模型是非马尔可夫模型。通过将霍克斯模型映射到马尔可夫随机偏微分方程上，我们开发了一种基于概率密度泛函的现场理论方法。通过求解稳态方程，我们可以预测接近临界点的强度的概率密度函数的幂律定标$ñ=\mathrm{1个}$ 霍克斯过程的指数，具有非通用指数，是背景强度的函数 ${\nu }_0$ 霍克斯强度，内存内核的平均时间尺度和分支比的关系 $ñ$。我们的理论预测已通过数值模拟得到证实。