{"id":3095,"date":"2011-09-24T11:04:41","date_gmt":"2011-09-24T09:04:41","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=3095"},"modified":"2014-06-17T17:32:54","modified_gmt":"2014-06-17T15:32:54","slug":"ten-years","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2011\/09\/24\/ten-years\/","title":{"rendered":"Ten years"},"content":{"rendered":"

\"Mathematician<\/a><\/p>\n

I have spent months working alone on a conjecture, ten years ago. It was at the end of my doctorate under the supervision of M. Ledoux<\/a>. The problem was as follows: let \\( {V:\\mathbb{R}\\rightarrow\\mathbb{R}} \\) be a smooth function of the form \\( {V=U+B} \\) with \\( {U} \\) uniformly convex and \\( {B} \\) bounded, such that<\/p>\n

\\[ \\int_{\\mathbb{R}}\\!e^{-V(t)}\\,dt=1. \\]<\/p>\n

For every integer \\( {n\\geq1} \\) and real \\( {m} \\), let \\( {\\mu_n} \\) be the product probability measure<\/p>\n

\\[ d\\mu_n=e^{-V(x_1)-\\cdots-V(x_n)}dx_1\\cdots dx_n \\]<\/p>\n

and let \\( {\\mu_{n,m}} \\) be the conditional exchangeable probability measure<\/p>\n

\\[ \\mu_{n,m}=\\mu(\\,\\cdot\\,|\\,x_1+\\cdots+x_n=m). \\]<\/p>\n

The conjecture stated that there exists a real constant \\( {C=C(U,B)} \\) such that for every \\( {n\\geq1} \\), \\( {m\\in\\mathbb{R}} \\), and every smooth function \\( {f:\\mathbb{R}^n\\rightarrow\\mathbb{R}} \\) in the unit sphere of \\( {L^2(\\mu_{n,m})} \\),<\/p>\n

\\[ \\int\\!f^2\\log(f^2)\\,d\\mu_{n,m} \\leq C\\int\\!\\nabla f\\cdot\\nabla f\\,d\\mu_{n,m}, \\]<\/p>\n

This is a logarithmic Sobolev inequality, stronger than the Poincar\u00e9 version of the conjecture which states that for every \\( {\\mu_{n,m}} \\)-centered smooth \\( {f:\\mathbb{R}^n\\rightarrow\\mathbb{R}} \\),<\/p>\n

\\[ \\int\\!f^2\\,d\\mu_{n,m} \\leq C\\int\\!\\nabla f\\cdot\\nabla f\\,d\\mu_{n,m}. \\]<\/p>\n

The main feature of the conjecture is the independence of \\( {C} \\) over the constant \\( {m} \\) and the dimension \\( {n} \\). An important step was made by Landim, Panizo, and Yau<\/a> for another right hand side, in the case where \\( {U} \\) is quadratic and \\( {B,B',B''} \\) are bounded, using the martingale decomposition method of Lu-Yau<\/a> or Stroock-Zegarlinski<\/a>. The Poincar\u00e9 version of the conjecture was almost solved by Caputo<\/a> using a projection technique borrowed from a work of Carlen, Carvalho, and Loss<\/a>. My modest contribution<\/a> consisted in a proof of the conjecture in the case where \\( {U} \\) is quadratic and \\( {B,B',B''} \\) are bounded, using the martingale technique. Later, Grunewald, Otto, Villani, and Westdickenberg<\/a> provided a new technique based on a two-scales decomposition (without improving the result at that time). One can also mention some contributions by Barthe and Wolff<\/a> for the Poincar\u00e9 case, and by Leli\u00e8vre<\/a>, among others.<\/p>\n

Recently, Otto and his PhD student Menz have solved<\/a> completely the conjecture, using the two-scales decomposition of Grunewald et al and a new one dimensional covariance estimate of the following form: for every smooth \\( {\\nu} \\)-centered \\( {f,g:\\mathbb{R}\\rightarrow\\mathbb{R}} \\) where \\( {d\\nu=e^{-U(t)}dt} \\),<\/p>\n

\\[ \\int\\!fg\\,d\\nu\\leq \\left\\Vert U''^{-1}g'\\right\\Vert_\\infty\\int\\!|f'|\\,d\\nu. \\]<\/p>\n

This new covariance estimate is a one dimensional asymmetric \\( {L^1-L^\\infty} \\) covariance version of the classical Brascamp-Lieb inequality. Some extensions of such covariance inequalities were obtained very recently by Carlen, Cordero-Erausquin, and Lieb<\/a>.<\/p>\n

The Brascamp-Lieb inequality states that if \\( {d\\nu=e^{-H(x)}\\,dx} \\) is a strictly log-concave probability measure on \\( {\\mathbb{R}^n} \\), then for every \\( {\\nu} \\)-centered smooth function \\( {f:\\mathbb{R}^n\\rightarrow\\mathbb{R}} \\),<\/p>\n

\\[ \\int\\!f^2\\,d\\nu \\leq \\int\\!\\nabla f\\cdot (\\nabla^2H)^{-1}\\nabla f\\,d\\nu. \\]<\/p>\n

The original proof of Brascamp and Lieb (1976) goes by induction on the dimension using the fact that the marginals of a log-concave probability measure are still log-concave. There is an alternative quick proof using the Witten Laplacian on closed forms. Namely, and formally, if \\( {f:\\mathbb{R}^n\\rightarrow\\mathbb{R}} \\) is a smooth function with zero \\( {\\nu} \\)-mean, then we may write \\( {f=L g} \\) where<\/p>\n

\\[ L=\\Delta-\\nabla H\\cdot\\nabla. \\]<\/p>\n

A this point, we recall the (Bochner) commutation formula<\/p>\n

\\[ \\nabla L=L\\nabla - (\\nabla^2 H)\\nabla=\\vec L\\nabla \\quad\\mbox{where}\\quad \\vec L=L-\\nabla^2H \\]<\/p>\n

The gradient of a smooth function is a closed \\( {1} \\)-form and \\( {\\vec L} \\) acts on such forms (Witten Laplacian on forms). Taking \\( {\\nabla g=\\vec{L}^{-1}\\nabla f} \\) and using integration by parts,<\/p>\n

\\[ \\int\\!f^2\\,d\\nu =\\int\\!f Lg\\,d\\nu =-\\int\\!\\nabla f\\cdot \\nabla g\\,d\\nu =-\\int\\!\\nabla f \\cdot \\vec{L}^{-1}\\nabla f\\,d\\nu. \\]<\/p>\n

Now, as quadratic forms, since \\( {-L\\geq0} \\) we get \\( {-\\vec L^{-1}\\leq (\\nabla^2 H)^{-1}} \\).<\/p>\n

This was explored by Helffer and Sj\u00f6strand<\/a>. It was also known by H\u00f6rmander<\/a> (according to Cordero-Erausquin). This approach, efficient for the variance, fails miserably for the covariance.<\/p>\n

A recent talk<\/a> by Villani provides an overview on this conjecture, its recent solution, and its relation with Ginzburg-Landau dynamics and hydrodynamical limits in statistical mechanics.<\/p>\n","protected":false},"excerpt":{"rendered":"

I have spent months working alone on a conjecture, ten years ago. It was at the end of my doctorate under the supervision of M.…<\/p>\n

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