  We derive the modified diffusion equations defined on kappa-space-time and
using these, investigate the change in the spectral dimension of
kappa-space-time with the probe scale. These deformed diffusion equations are
derived by applying Wick's rotation to the $\kappa$-deformed
Schr$\ddot{o}$dinger equations obtained from different choices of Klein-Gordon
equations in the $\kappa$-deformed space-time. Using the solutions of these
equations, obtained by perturbative method, we calculate the spectral dimension
for different choices of the generalized Laplacian and analyse the dimensional
flow in the $\kappa$-space-time. In the limit of commutative space-time, we
recover the well known equality of spectral dimension and topological
dimension. We show that the higher derivative term in the deformed diffusion
equations make the spectral dimension unbounded (from below) at high energies.
We show that the finite mass of the probe results in the spectral dimension to
become infinitely negative at low energies also. In all the cases, we have
analysed the effect of finite size of the probe on the spectral dimension.
