On the distribution of inhomogeneous functionals of Brownian local time at the inverse local time moment

We consider the question: how to calculate distributions of the simplest inhomogeneous integral functional of Brownian local time with respect to space parameter at the inverse local time moment. For the Laplace transform of distribution of such a functional we obtain 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 considered at the inverse local time moment are derived.