Time-dependent weak rate of convergence for functions of generalized bounded variation

Let $W$ denote the Brownian motion. For any exponentially bounded Borel function $g$ the function $u$ defined by $u(t,x)= $\mathbb{E}$[g(x{+}\sigma W_{T-t})]$ is the stochastic solution of the backward heat equation with terminal condition $g$. Let $u^n(t,x)$ denote the corresponding approximation generated by a simple symmetric random walk with time steps $2T\mathrm{/}n$ and space steps $\pm \sigma \sqrt{T\mathrm{/}n}$ where $\sigma >0$. For quite irregular terminal conditions $g$ (bounded variation on compact intervals, locally H"older continuous) the rate of convergence of $u^n(t,x)$ to $u(t,x)$ is considered, and also the behavior of the error $u^n(t,x)-u(t,x)$ as $t$ tends to $T$