Exemplar-based texture synthesis consists in producing new synthetic images which have the same perceptual characteristics than a given texture sample while exhibiting sufficient innovation (to avoid verbatim copy). In this paper, we propose to address this problem with a model obtained as local transformations of Gaussian random fields. The local transformations operate on 3x3 patches and are designed to solve a semi-discrete optimal transport problem in order to reimpose the patch distribution of the exemplar texture. The semi-discrete optimal transport problem is solved with a stochastic gradient algorithm, whose convergence speed is evaluated on several practical transport cases.After studying the properties of such transformed Gaussian random fields, we propose a multiscale extension of the model which aims at preserving the patch distribution of the exemplar texture at multiple scales. Experiments demonstrate that this multiscale model is able to synthesize structured textures while keeping several mathematical guarantees, and with low requirements in synthesis time and memory storage. In particular, a single patch optimal transport map is shown to be better than iterated nearest neighbor assignments in terms of statistical guarantees. Besides, once the model is estimated, the resulting synthesis algorithm is fast and highly parallel since it amounts to perform weighted nearest neighbor patch assignments at each scale.