  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 relies on the Hadamard's variational formula. Namely, the variation of
pressure in stochastic Laplacian growth near the interface is given by the
covariance of martingales on the double.
