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We construct a family of non-parametric (infinite-dimensional) manifolds offinite measures on $R^d$. The manifolds are modelled on a variety of weightedSobolev spaces, including Hilbert-Sobolev spaces and mixed-norm spaces. Eachsupports the Fisher-Rao metric as a weak Riemannian metric. Densities areexpressed in terms of a deformed exponential function having linear growth.Unusually for the Sobolev context, and as a consequence of its linear growth,this "lifts" to a nonlinear superposition (Nemytskii) operator that actscontinuously on a particular class of mixed-norm model spaces, and on the fixednorm space $W^{2,1}$; i.e. it maps each of these spaces continuously intoitself. It also maps continuously between other fixed-norm spaces with a lossof Lebesgue exponent that increases with the number of derivatives. Some of theresults make essential use of a log-Sobolev embedding theorem. Each manifoldcontains a smoothly embedded submanifold of probability measures. Applicationsto the stochastic partial differential equations of nonlinear filtering (andhence to the Fokker-Planck equation) are outlined.

A Class of Non-Parametric Statistical Manifolds modelled on Sobolev Space