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id="measure_theory">Measure theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/measurable+space">measurable space</a>, <a class="existingWikiWord" href="/nlab/show/measurable+locale">measurable locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measure">measure</a>, <a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+measure+theory">geometric measure theory</a></p> </li> </ul> <h2 id="probability_theory">Probability theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+space">probability space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+distribution">probability distribution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state">state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/states+in+AQFT+and+operator+algebra">in AQFT and operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entropy">entropy</a>, <a class="existingWikiWord" href="/nlab/show/relative+entropy">relative entropy</a></p> </li> </ul> <h2 id="information_geometry">Information geometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/information+geometry">information geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/information+metric">information metric</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wasserstein+metric">Wasserstein metric</a></p> </li> </ul> <h2 id="thermodynamics">Thermodynamics</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second+law+of+thermodynamics">second law of thermodynamics</a>, <a class="existingWikiWord" href="/nlab/show/generalized+second+law+of+theormodynamics">generalized second law</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ergodic+theory">ergodic theory</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riesz+representation+theorem">Riesz representation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Finetti%27s+theorem">de Finetti's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/law+of+large+numbers">law of large numbers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+extension+theorem">Kolmogorov extension theorem</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/machine+learning">machine learning</a>, <a class="existingWikiWord" href="/nlab/show/neural+networks">neural networks</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#references_and_links'>References and links</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>Wasserstein metric</em> is a certain <a class="existingWikiWord" href="/nlab/show/metric">metric</a> over a space of <a class="existingWikiWord" href="/nlab/show/probability+measure">probability measure</a>s on a <a class="existingWikiWord" href="/nlab/show/measurable+space">measurable space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>By (<a href="#JKO">JKO</a>) the <span class="newWikiWord">heat flow<a href="/nlab/new/heat+flow">?</a></span>/<span class="newWikiWord">diffusion equation<a href="/nlab/new/diffusion+equation">?</a></span> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/gradient+flow">gradient flow</a> of the <a class="existingWikiWord" href="/nlab/show/entropy">Boltzman-Shannon entropy</a> functional with respect to the Wasserstein metric.</p> <p>The Wasserstein metric does not seem to arise from a <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemann metric tensor</a>. A detailed discussion of the relevant gradient flows in non-smooth <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>s is in (<a href="#AGS">AGS</a>).</p> <h2 id="references_and_links">References and links</h2> <ul> <li>Related concepts <a class="existingWikiWord" href="/nlab/show/information+metric">information metric</a>, <a class="existingWikiWord" href="/nlab/show/entropy">entropy</a></li> </ul> <p>The characterization of heat flow as the gradient flow of Shannon-entropy is due to</p> <ul id="JKO"> <li>R. Jordan, D. Kinderlehrer, F. Otto, <em>The variational formulation of the <a class="existingWikiWord" href="/nlab/show/Fokker-Planck+equation">Fokker-Planck equation</a></em> , SIAM J. Math. Anal. 29 (1998), no. 1, 1-17.(<a href="http://www.imati.cnr.it/~savare/Ravello2010/JKO.pdf">pdf</a>)</li> </ul> <p>The analog of this for finite probability spaces is discussed in</p> <ul> <li>Jan Maas, <em>Gradient flows of the entropy for finite Markov chains</em> (<a href="http://www.janmaas.org/papers/discrete.pdf">pdf</a>)</li> </ul> <p>A comprehensive discussion of the corresponding gradient flows is in</p> <ul> <li>l Ambrosio, N. Gigli, G. Savaré, <em>Gradient flows in metric spaces and in the space of probability measures</em>, Second edition. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008. x+334 pp. (<a href="http://www.imati.cnr.it/~savare/research/optimal_transportation/Introduction.pdf">pdf</a> of toc and introduction) <a href="http://www.ams.org/mathscinet-getitem?mr=2401600">MR2009h:49002</a></li> <li>Luigi Ambrosio, Nicola Gigli, <em>Construction of the parallel transport in the Wasserstein space</em>, Methods Appl. Anal. 15 (2008), no. 1, 1–29, <a href="http://www.ams.org/mathscinet-getitem?mr=2482206">MR2010c:49082</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 26, 2019 at 01:44:38. 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