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Description
We introduce a diffusive transport metric between probability measures that generalizes the Hellinger, Kantorovich and martingale transport.
We represent several classes of parabolic PDEs as metric gradient flows with respect to diffusive transport, such as linear second-order diffusion equations in non-divergence (Itô) form, the quadratic porous medium equation, and the fourth-order DLSS equation.
We propose a spatial discretization of the fourth-order nonlinear DLSS equation on the circle and the line, based on its diffusive gradient flow structure. The discrete dynamics inherits this gradient flow structure, and additionally satisfies contractivity in the Hellinger distance and monotonicity of several Lyapunov functionals. We prove convergence of the scheme as the mesh size vanishes, and exponential convergence in time for the equation rewritten in self-similar variables.
Based on joint works with Daniel Matthes, Eva-Maria Rott, Giuseppe Savaré.