The motion of a quantum particle in the presence of the washboard potential and linear friction is studied. We find the general expressions for the occupation probabilities in the quasi-momentum (quasi-charge) space and prove their duality to the analogous quantities in the coordinate space. Quantum coherence destruction, diffusion and localization in the quasi-momentum space are discussed and the mobility of the particle is calculated. The influence of quantum effects on the I- V curves of small Josephson junctions is examined. It is shown that the region of Bloch oscillations is restricted by the condition R>R Q=π/2e 2≈6.5 kΩ , while Josephson oscillations exist at R< RQ ( R being the shunt resistance).