 Solution

Let $\rho =t\wedge {\tau }_0\wedge {\tau }_1$ and apply the Itô formula to $u(t-s,B_s)$ for $0\le s\le \rho$. The heat equation cancels the drift, so the stopped process is a bounded martingale. The optional sampling theorem for a supermartingale gives $u(t,x)={\mathbb{E}}_x[u(t-\rho ,B_{\rho })]$. On the three mutually exclusive terminal events, the initial and Dirichlet boundary conditions identify this value asThis is the probabilistic representation of the heat equation with time-dependent Dirichlet data.