  The study of multidimensional stochastic processes involves complex
computations in intricate functional spaces. In particular, the diffusion
processes, which include the practically important Gauss-Markov processes, are
ordinarily defined through the theory of stochastic integration. Here, inspired
by the L\'{e}vy-Cieselski construction of the Wiener process, we propose an
alternative representation of multidimensional Gauss-Markov processes as
expansions on well-chosen Schauder bases, with independent random coefficients
of normal law with zero mean and unitary variance. We thereby offer a natural
multi-resolution description of Gauss-Markov processes as limits of the
finite-dimensional partial sums of the expansion, that are strongly
almost-surely convergent. Moreover, such finite-dimensional random processes
constitute an optimal approximation of the process, in the sense of minimizing
the associated Dirichlet energy under interpolating constraints. This approach
allows simpler treatment in many applied and theoretical fields and we provide
a short overview of applications we are currently developing.
