A prediction problem for the Brownian sheet

The information given by the position of the Brownian sheet along or near a curve can be represented by the sharp field, the minimal splitting field, or the germ field. When the curve is a separation line, we show that the last two fields are always equal and give necessary and sufficient conditions for equality of the first and third. Through explicit integral expressions for the conditional expectation of a Gaussian random variable with respect to the germ and sharp fields, we show that the germ field generally gives a second-order predictor of the position of the Brownian sheet, whereas the sharp field only gives a first-order one.