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For a regular transient diffusion, we provide a decomposition of its lastpassage time to a certain state $\alpha$. This is accomplished by transformingthe original diffusion into two diffusions using the occupation time of thearea above and below $\alpha$. Based on these two processes, the decompositionformula of the Laplace transform of the last passage time is derived explicitlyin a simple form in terms of Green functions. This formula may be used forfurther investigations of diffusions with switching parameters. As one example,we demonstrate an application to a diffusion with two-valued drift.

On Decomposition of the Last Passage Time of Diffusions