A family of exponential martingales of a stochastic Laplacian growth problem is proposed. Stochastic Laplacian growth describes a regularized interface dynamics in a two-fluid system, where the viscous fluid is incompressible at a large scale, while compressible at a small scale in the vicinity of the interface. Hence, random fluctuations of pressure near the boundary are inevitable. By using Loewner–Kufarev equation, we study interface dynamics generated by nonlocal random Loewner measure, which produces the patterns with viscous fingers. We use a Schottky double construction to introduce a one-parametric family of functions of random processes on the double closely connected to the correlation functions of primary operators of the boundary conformal field theory in the Coulomb gas framework. For a specific value of the parameter, these functions are martingales with respect to stochastic Loewner flow on the Schottky double. A connection between the proposed algebraic construction and the physical problem of stochastic interface dynamics in the Hele-Shaw cell relies on the Hadamard’s variational formula. Namely, the variation of pressure in stochastic Laplacian growth in the vicinity of the interface is given by the variance of martingales on the double.

提出了一个随机拉普拉斯增长问题的指数mar族。随机拉普拉斯算子的增长描述了在双流体系统中规则的界面动力学，其中粘性流体在大范围内不可压缩，而在界面附近小范围内可压缩。因此，边界附近的压力的随机波动是不可避免的。通过使用Loewner–Kufarev方程，我们研究了由非局部随机Loewner测度生成的界面动力学，该界面动力学产生带有粘性手指的图案。我们使用肖特基对偶构造在与库伦气体框架中边界共形场理论的主算符紧密相关的对偶上引入随机过程的一参数函数族。对于参数的特定值，这些功能相对于肖特基双上的随机Loewner流是mar。拟议的代数构造与Hele-Shaw单元中随机界面动力学的物理问题之间的联系依赖于Hadamard的变分公式。即，界面上随机拉普拉斯算子生长压力的变化是由the上的变化决定的。