Wheeler-De Witt equation for brane gravity

We consider gravity in the system consisting of the Bogomol'nyi-Prasad-Sommerfield (BPS) $D3$-brane embedded in a flat background geometry, produced by the solutions of supergravity. The effective action for this system is represented by the sum of the Hilbert-Einstein and Dirac-Born-Infeld actions. We derive the Wheeler-DeWitt equation for this system and obtain analytical solutions in some special cases. We also calculate the tunneling probability from the Planckian size of the $D3$-brane to the classical regime.