Abstract
We introduce the transportation-annihilation distanceW_p^sharp between subprobabilities and derive contraction estimates with respect to this distance for the heat flow with homogeneous Dirichlet boundary conditions on an open set in a metric measure space. We also deduce the Bochner inequality for the Dirichlet Laplacian as well as gradient estimates for the associated Dirichlet heat flow. For the Dirichlet heat flow, moreover, we establish a gradient flow interpretation within a suitable space of charged probabilities. In order to prove this, we will work with the doubling of the open set, the space obtained by gluing together two copies of it along the boundary.
Highlights
Introduction and statement of main resultsWe present an approach to heat flow with homogeneous Dirichlet boundary conditions via optimal transport—the very first ever—based on a novel particle interpretation for this evolution
The RCD(K, ∞)-condition means that the metric measure space (X, d, m) is infinitesimally Hilbertian with Ricci curvature bounded from below by K in the sense of Lott-Sturm-Villani, [13,24]
Remark 1.18 (a) Note that, due to the isometric embedding of P2(X ) into P2(Y |X ), this assumption will imply the K -convexity of Entm in P2(X ), W2 and the CD(K, ∞)condition for the metric measure space (X, d, m)
Summary
We present an approach to heat flow with homogeneous Dirichlet boundary conditions via optimal transport—the very first ever—based on a novel particle interpretation for this evolution. Our new interpretation will be based on particles moving around in Y , which are reflected if they hit the boundary, and which thereby randomly change their “charge”: half of them change into “antiparticles”, half of them continue to be normal particles. They annihilate each other but the total number of charged particles remains constant. We will interpret the pairs of subprobability measures (σ +, σ −) as a probability measure on the doubling of Y in X , i.e. a space obtained by gluing together two copies of X along the ”boundary“ X \ Y Laplacian on a convex subset of a Riemannian manifold—which surprisingly involves both, the Dirichlet Laplacian and the Neumann Laplacian
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