Functional Large Deviations for Kac–Stroock Approximation to a Class of Gaussian Processes with Application to Small Noise Diffusions

In this paper, we establish the functional large deviation principle (LDP) for the Kac–Stroock approximations of a wild class of Gaussian processes constructed by telegraph types of integrals with \(L^2\)-integrands under mild conditions and find the explicit form for their rate functions. Our investigation includes a broad range of kernels, such as those related to Brownian motions, fractional Brownian motions with whole Hurst parameters, and Ornstein–Uhlenbeck processes. Furthermore, we consider a class of non-Markovian stochastic differential equations driven by the Kac–Stroock approximation and establish their Freidlin–Wentzell type LDP. The rate function clearly indicates an interesting phase transition phenomenon as the approximating rate moves from one region to the other.