Transport and diffusion of plasma particles perpendicular and parallel to the magnetic field is discussed in the framework of the transverse drift wave theory. The starting model includes the density and magnetic field gradients perpendicular to the magnetic field vector. In such an inhomogeneous environment the transverse drift wave naturally develops. The transverse drift wave is a low frequency mode, with the frequency far below the ion gyro-frequency, it is driven by these gradients and it propagates perpendicular to them. The mode is also purely perpendicular to the magnetic field and it is electromagnetically transverse, which implies that when its wave vector is perpendicular to the magnetic field vector, the perturbed electric field is along the equilibrium magnetic field, while in the same time the perturbed magnetic field is in the direction of the background gradients. In application to the solar wind, it is shown that very small wave electric field amplitude, of the order of 10-7 V/m, within one wave period can produce the drift of protons in both directions, perpendicular to the ecliptic plane and also along the background magnetic field, to distances measured in millions of kilometers. The electric field along the magnetic field vector implies particle acceleration in the same direction. When a critical threshold velocity of the particle is achieved, the particle motion becomes stochastic. This is a completely new nonlinear stochastic mechanism which follows from the very specific geometry of the transverse drift mode. Particle drift perpendicular to the magnetic field vector means a diffusion of particles, with the effective diffusion coefficient for ions that is at least 11 orders of magnitude larger than the classic diffusion coefficient. The features of this diffusion are: within certain time interval, initially faster particles will diffuse to larger distances, and the same holds for protons in comparison to heavier ions. For electrons the effective diffusion coefficient can easily match the one obtained from observations, i.e., to become of the order of 1017 m2/s. It is also expected that the wave-induced stochastic motion will considerably increase the effective collision frequency in such an environment which is, with respect to its mean parameters, practically collision-lees. Hence, the solar wind regions affected by such a stochastic acceleration may show various unexpected features that are typical for collisional plasmas.