On the distribution of inhomogeneous functionals of Brownian local time

We consider the question: how to calculate distributions of the simplest inhomogeneous integral functional of Brownian local time with respect to space parameter. For the Laplace transform of distribution of such a functional we obtaine formulas expressed in terms of solutions of the second order differential equations, satisfying some boundary conditions. As an application of these formulas the joint distribution of suprema of Brownian local time at adjacent intervals are derived.