On Small Deviation Asymptotics In L2 of Some Mixed Gaussian Processes

We study the exact small deviation asymptotics with respect to the Hilbert norm for some mixed Gaussian processes. The simplest example here is the linear combination of the Wiener process and the Brownian bridge. We get the precise final result in this case and in some examples of more complicated processes of similar structure. The proof is based on Karhunen-Loève expansion together with spectral asymptotics of differential operators and complex analysis methods.