English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Since the early nineties, it has been observed that the Schroedinger bridgeproblem can be formulated as a stochastic control problem with atypicalboundary constraints. This in turn has a fluid dynamic counterpart where theflow of probability densities represents an entropic interpolation between thegiven initial and final marginals. In the zero noise limit, such entropicinterpolation converges in a suitable sense to the displacement interpolationof optimal mass transport (OMT). We consider two absolutely continuous curvesin Wasserstein space ${\cal W}_2$ and study the evolution of the relativeentropy on ${\cal W}_2\times {\cal W}_2$ on a finite time interval. Thus, thisstudy differs from previous work in OMT theory concerning relative entropy froma fixed (often equilibrium) distribution (density). We derive a gradient flowon Wasserstein product space. We find the remarkable property that fluxes inthe two components are opposite. Plugging in the "steepest descent" into theevolution of the relative entropy we get what appears to be a new formula: Thetwo flows approach each other at a faster rate than that of two solutions ofthe same Fokker-Planck. We then study the evolution of relative entropy in thecase of uncontrolled-controlled diffusions. In two special cases of theSchroedinger bridge problem, we show that such relative entropy may bemonotonically decreasing or monotonically increasing.

Stochastic control, entropic interpolation and gradient flows on Wasserstein product spaces