Let us consider a moving point $x_k\in \mathbb{R}$, motion of which follows a gaussian random walk: \begin{equation} x_{k+1} = x_{k} + w_k,\;\;w_{k}\sim\mathcal{N}(0, \sigma). \end{equation} What I want to do is alogirithmically sample sequences $\{x_k\}_{k=0}^T$ which satisfies $x_0=x_T=0$. Does anyone know how to do this?

Note: A naive way occured to my mind was sampling many randomwalk realizations starting from $x_0=0$ and pick up ones that satisfy the $|x_T|<\varepsilon$, where $\varepsilon$ is a small value and is a tuning parameter. But this is not what I am talking about.


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