Karhunen-Loeve expansions of alpha-Wiener bridges

We study Karhunen-Loeve expansions of the process $(X_t^{(\alpha)})_{t\in[0,T)}$ given by the stochastic differential equation $dX_t^{(\alpha)} = -\frac\alpha{T-t} X_t^{(\alpha)} dt+ dB_t,$ $t\in[0,T),$ with an initial condition $X_0^{(\alpha)}=0,$ where $\alpha>0,$ $T\in(0,\infty)$ and $(B_t)_{t\geq 0}$ is a standard Wiener process. This process is called an $\alpha$-Wiener bridge or a scaled Brownian bridge, and in the special case of $\alpha=1$ the usual Wiener bridge. We present weighted and unweighted Karhunen-Loeve expansions of $X^{(\alpha)}$. As applications, we calculate the Laplace transform and the distribution function of the $L^2[0,T]$-norm square of $X^{(\alpha)}$ studying also its asymptotic behavior (large and small deviation).