  The dynamical evolution of a recently introduced one dimensional model in
\cite{biswas-sen} (henceforth referred to as model I), has been made stochastic
by introducing a parameter $\beta$ such that $\beta =0$ corresponds to the
Ising model and $\beta \to \infty$ to the original model I. The equilibrium
behaviour for any value of $\beta $ is identical: a homogeneous state. We
argue, from the behaviour of the dynamical exponent $z$,that for any $\beta
\neq 0$, the system belongs to the dynamical class of model I indicating a
dynamic phase transition at $\beta = 0$. On the other hand, the persistence
probabilities in a system of $L$ spins saturate at a value $P_{sat}(\beta, L) =
(\beta/L)^{\alpha}f(\beta)$, where $\alpha$ remains constant for all $\beta
\neq 0$ supporting the existence of the dynamic phase transition at $\beta =0$.
The scaling function $f(\beta)$ shows a crossover behaviour with $f(\beta) =
\rm{constant} $ for $\beta <<1$ and $f(\beta) \propto \beta^{-\alpha}$ for
$\beta >>1$.
