We study the kinetics of protein folding via statistical energy landscape theory . In particular, we concentrate on the local-connectivity case, where conformation change or transition can only happen between neighboring states with folding progress described in terms of an order parameter given by fraction of native conformation. Diffusion dynamics is analyzed in detail and an expression for mean first passage time (MFPT) from non-native unfolded states to native folded state is obtained. It was found that the MFPT has a inverted bell-like curve dependent on the temperature. At high temperature, the MFPT is longer due to the instability of the native folded state. At low temperature, the MFPT is also longer due to the possible trapping into low lying non-native folded states. We find that the MFPT is shorter when the ratio of the spread of the folding energy landscape versus the energy gap is increased. The fluctuation variance and higher-order moments are studied to infer the distribution of the first passage time (FPT). At high temperature, the distribution becomes close to a Poisson distribution, while at low temperatures the distribution becomes close to the L\'evy distribution developing fatty tails, indicating the rare event might give great contribution. The intermittency phonomena or the non-self averaging character of the folding dynamics emerges. The application of the results to single molecule dynamics experiments, where a power law (L\'evy) distribution of relaxation time of the underline protein energy landscape is observed.