The integral equation is T(x_3,t_3|x_1,t_1)=\int \text{d}x_2T(x_3,t_3|x_2,t_2)T(x_2,t_2|x_1,t_1) where T(x_3,t_3|x_1,t_1) is the probability density of a Markov process taking the value x_3 at time t_3 given that it took the value of x_1 at time t_1. So far so good. To derive the differential...