By Luigi Ambrosio
This e-book is dedicated to a thought of gradient flows in areas which aren't inevitably endowed with a usual linear or differentiable constitution. It involves elements, the 1st one pertaining to gradient flows in metric areas and the second dedicated to gradient flows within the area of chance measures on a separable Hilbert area, endowed with the Kantorovich-Rubinstein-Wasserstein distance.
The elements have a few connections, when you consider that the distance of likelihood measures presents an enormous version to which the "metric" conception applies, however the publication is conceived in one of these manner that the 2 elements should be learn independently, the 1st one by means of the reader extra attracted to non-smooth research and research in metric areas, and the second via the reader extra oriented in the direction of the functions in partial differential equations, degree thought and probability.
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Additional info for Gradient flows: In metric spaces and in the space of probability measures
Sample text
19). Even in a metric framework, if a curve of maximal slope is a Generalized Minimizing Movement, it exhibits a sort of regularizing effect allowing for a finer description of the differential equation at each point of the interval, if we consider right derivatives. 2, page. 57]. 15. 5 for some λ ∈ R.
5, |∂φ|(v) is a strong upper gradient. Proof. 19) are simple consequence of the monotonicity of difference quotients of convex functions. 17). 16), which yields |φ(v) − φ(vn )| ≤ vn − v ξn ∗ + ξ ∗ . 36 Chapter 1. Curves and Gradients in Metric Spaces The inequality w φ(v) − φ(v + w) ≤ ξ, w w ∀w ∈ B \ {0} yields that lφ (v) can be estimated from above by ξ B for any ξ ∈ ∂φ. Assuming that lφ (v) is finite, to conclude the proof we need only to show the existence of ξ ∈ ∂φ(v) such that ξ B ≤ lφ (v).
Proof. 7) and the differentiability of φ2 . 17) being φ2 continuous. 4. Curves of maximal slope in Hilbert and Banach spaces 37 that Dφ2 (v) = 0; it follows that φ(w) − φ(v) |∂φ|(v) = lim sup w−v w→v + + φ1 (w) − φ1 (v) |φ2 (w) − φ2 (w)| − lim sup w−v w−v w→v w→v ◦ ◦ = |∂φ1 |(v) = ∂ φ1 (v) ∗ = ∂ φ(v) ∗ . 9), we conclude. Let us rephrase the last conclusion of the previous Corollary, which is quite interesting in the case B does not satisfy the Radon-Nikod´ ym property (as in many examples of rate-independent problems, see [114, 113]).