  We introduce the analogue of Dunkl processes in the case of an affine root
system of type $\widetilde{\text{A}}_1$. The construction of the affine Dunkl
process is achieved by a skew-product decomposition by means of its radial part
and a jump process on the affine Weyl group, where the radial part of the
affine Dunkl process is defined as the unique solution of some stochastic
differential equation. We prove that the affine Dunkl process is a c\`adl\`ag
Markov process as well as a local martingale, study its jumps, and give a
martingale decomposition, which are properties similar to those of the
classical Dunkl process.
