A stochastic telegraph equation is defined by adding a random inhomogeneity to the classical (second-order linear hyperbolic) telegraph differential equation. The inhomogeneities we consider are proportional to the two-dimensional white noise, and solutions to our equation are two-dimensional random Gaussian fields. We show that such fields arise naturally as asymptotic fluctuations of the height function in a certain limit regime of the stochastic six-vertex model in a quadrant. The corresponding law of large numbers—the limit shape of the height function—is described by the (deterministic) homogeneous telegraph equation.
"A stochastic telegraph equation from the six-vertex model." Ann. Probab. 47 (6) 4137 - 4194, November 2019. https://doi.org/10.1214/19-AOP1356