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Otis Chodosh
Otis Chodosh
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von Renesse and Sturm have constructed a family "natural" measures $\mu$ making $(\mathcal{P}([0,1]),d_{W_2},\mu_\beta)$ into a metric measure space: http://www.ams.org/mathscinet-getitem?mr=2537551. They argue that their measure can be formally thought of as $$ \tag{*} \mu_\beta = C_\beta^-1 e^{-\beta Ent(\cdot | \text{Leb})} \mu_{unif} $$$$ \tag{*} \mu_\beta = C_\beta^{-1} e^{-\beta Ent(\cdot | \text{Leb})} \mu_{unif} $$ where $C_\beta$ is a normalization constant, $Ent(\rho \text{Leb} |\text{Leb}) = \int_{[0,1]}\rho \log \rho$ (and is infinite for non-absolutely continuous measures), and $\mu_{unif}$ is a sort of "uniform" measure.

Sturm later extended this to allow the underling space to be any $n$-dimensional Riemannian manifold: http://www.ams.org/mathscinet-getitem?mr=2857031.

However, (*) is not completely rigorous, and the measures do not behave quite as one would expect. There is a formal infinite dimensional Riemannian structure on the regular part of $(\mathcal{P}(M),d_{W_2})$ (i.e. on the subset of measures of the form $\rho dV_g$ where $\rho \in C^\infty(M)$) that goes back to Otto (http://www.ams.org/mathscinet-getitem?mr=1842429). One might hope that this Riemannian structure (formally) induced some sort of measure, and that one could use this formal argument to find an actual measure on $\mathcal{P}(M)$, which behaves nicely. However, it seems like while a good notion of "gradient" exists in this formal setting, a Laplacian does not. This suggests that there is no natural measure because otherwise the Laplacian could be defined by integration by parts. See Gigli's monograph http://www.ams.org/mathscinet-getitem?mr=2920736 (remark 5.6).

An obvious question given a measure on $(\mathcal{P}(M),d_{W_2})$ is whether or not it has lower Ricci curvature bounds in the sense of Lott--Villani--Sturm. When $M$ is the interval, this is not true: http://www.ams.org/mathscinet-getitem?mr=2900478. See also Section 8.3 in http://arxiv.org/pdf/1105.2883.pdf

In a slightly different direction. Erbar and Huesmann have recently studied http://link.springer.com/article/10.1007/s00526-014-0790-1 the curvature of the "configuration space" of a manifold and shown that there, there is a natural choice of measure which inherits lower Ricci curvature bounds from the underlying manifold. The configuration space is quite similar to the space of all probability measures (for example, the set of finite sums of dirac measures is dense in the space of probability measures with respect to weak convergence).

von Renesse and Sturm have constructed a family "natural" measures $\mu$ making $(\mathcal{P}([0,1]),d_{W_2},\mu_\beta)$ into a metric measure space: http://www.ams.org/mathscinet-getitem?mr=2537551. They argue that their measure can be formally thought of as $$ \tag{*} \mu_\beta = C_\beta^-1 e^{-\beta Ent(\cdot | \text{Leb})} \mu_{unif} $$ where $C_\beta$ is a normalization constant, $Ent(\rho \text{Leb} |\text{Leb}) = \int_{[0,1]}\rho \log \rho$ (and is infinite for non-absolutely continuous measures), and $\mu_{unif}$ is a sort of "uniform" measure.

Sturm later extended this to allow the underling space to be any $n$-dimensional Riemannian manifold: http://www.ams.org/mathscinet-getitem?mr=2857031.

However, (*) is not completely rigorous, and the measures do not behave quite as one would expect. There is a formal infinite dimensional Riemannian structure on the regular part of $(\mathcal{P}(M),d_{W_2})$ (i.e. on the subset of measures of the form $\rho dV_g$ where $\rho \in C^\infty(M)$) that goes back to Otto (http://www.ams.org/mathscinet-getitem?mr=1842429). One might hope that this Riemannian structure (formally) induced some sort of measure, and that one could use this formal argument to find an actual measure on $\mathcal{P}(M)$, which behaves nicely. However, it seems like while a good notion of "gradient" exists in this formal setting, a Laplacian does not. This suggests that there is no natural measure because otherwise the Laplacian could be defined by integration by parts. See Gigli's monograph http://www.ams.org/mathscinet-getitem?mr=2920736 (remark 5.6).

An obvious question given a measure on $(\mathcal{P}(M),d_{W_2})$ is whether or not it has lower Ricci curvature bounds in the sense of Lott--Villani--Sturm. When $M$ is the interval, this is not true: http://www.ams.org/mathscinet-getitem?mr=2900478. See also Section 8.3 in http://arxiv.org/pdf/1105.2883.pdf

In a slightly different direction. Erbar and Huesmann have recently studied http://link.springer.com/article/10.1007/s00526-014-0790-1 the curvature of the "configuration space" of a manifold and shown that there, there is a natural choice of measure which inherits lower Ricci curvature bounds from the underlying manifold. The configuration space is quite similar to the space of all probability measures (for example, the set of finite sums of dirac measures is dense in the space of probability measures with respect to weak convergence).

von Renesse and Sturm have constructed a family "natural" measures $\mu$ making $(\mathcal{P}([0,1]),d_{W_2},\mu_\beta)$ into a metric measure space: http://www.ams.org/mathscinet-getitem?mr=2537551. They argue that their measure can be formally thought of as $$ \tag{*} \mu_\beta = C_\beta^{-1} e^{-\beta Ent(\cdot | \text{Leb})} \mu_{unif} $$ where $C_\beta$ is a normalization constant, $Ent(\rho \text{Leb} |\text{Leb}) = \int_{[0,1]}\rho \log \rho$ (and is infinite for non-absolutely continuous measures), and $\mu_{unif}$ is a sort of "uniform" measure.

Sturm later extended this to allow the underling space to be any $n$-dimensional Riemannian manifold: http://www.ams.org/mathscinet-getitem?mr=2857031.

However, (*) is not completely rigorous, and the measures do not behave quite as one would expect. There is a formal infinite dimensional Riemannian structure on the regular part of $(\mathcal{P}(M),d_{W_2})$ (i.e. on the subset of measures of the form $\rho dV_g$ where $\rho \in C^\infty(M)$) that goes back to Otto (http://www.ams.org/mathscinet-getitem?mr=1842429). One might hope that this Riemannian structure (formally) induced some sort of measure, and that one could use this formal argument to find an actual measure on $\mathcal{P}(M)$, which behaves nicely. However, it seems like while a good notion of "gradient" exists in this formal setting, a Laplacian does not. This suggests that there is no natural measure because otherwise the Laplacian could be defined by integration by parts. See Gigli's monograph http://www.ams.org/mathscinet-getitem?mr=2920736 (remark 5.6).

An obvious question given a measure on $(\mathcal{P}(M),d_{W_2})$ is whether or not it has lower Ricci curvature bounds in the sense of Lott--Villani--Sturm. When $M$ is the interval, this is not true: http://www.ams.org/mathscinet-getitem?mr=2900478. See also Section 8.3 in http://arxiv.org/pdf/1105.2883.pdf

In a slightly different direction. Erbar and Huesmann have recently studied http://link.springer.com/article/10.1007/s00526-014-0790-1 the curvature of the "configuration space" of a manifold and shown that there, there is a natural choice of measure which inherits lower Ricci curvature bounds from the underlying manifold. The configuration space is quite similar to the space of all probability measures (for example, the set of finite sums of dirac measures is dense in the space of probability measures with respect to weak convergence).

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Otis Chodosh
Otis Chodosh
  • 7.2k
  • 3
  • 25
  • 56

von Renesse and Sturm have constructed a family "natural" measures $\mu$ making $(\mathcal{P}([0,1]),d_{W_2},\mu_\beta)$ into a metric measure space: http://www.ams.org/mathscinet-getitem?mr=2537551. They argue that their measure can be formally thought of as $$ \tag{*} \mu_\beta = C_\beta^-1 e^{-\beta Ent(\cdot | \text{Leb})} \mu_{unif} $$ where $C_\beta$ is a normalization constant, $Ent(\rho \text{Leb} |\text{Leb}) = \int_{[0,1]}\rho \log \rho$ (and is infinite for non-absolutely continuous measures), and $\mu_{unif}$ is a sort of "uniform" measure.

Sturm later extended this to allow the underling space to be any $n$-dimensional Riemannian manifold: http://www.ams.org/mathscinet-getitem?mr=2857031.

However, (*) is not completely rigorous, and the measures do not behave quite as one would expect. There is a formal infinite dimensional Riemannian structure on the regular part of $(\mathcal{P}(M),d_{W_2})$ (i.e. on the subset of measures of the form $\rho dV_g$ where $\rho \in C^\infty(M)$) that goes back to Otto (http://www.ams.org/mathscinet-getitem?mr=1842429). One might hope that this Riemannian structure (formally) induced some sort of measure, and that one could use this formal argument to find an actual measure on $\mathcal{P}(M)$, which behaves nicely. However, it seems like while a good notion of "gradient" exists in this formal setting, a Laplacian does not. This suggests that there is no natural measure because otherwise the Laplacian could be defined by integration by parts. See Gigli's monograph http://www.ams.org/mathscinet-getitem?mr=2920736 (remark 5.6).

An obvious question given a measure on $(\mathcal{P}(M),d_{W_2})$ is whether or not it has lower Ricci curvature bounds in the sense of Lott--Villani--Sturm. When $M$ is the interval, this is not true: http://www.ams.org/mathscinet-getitem?mr=2900478. See also Section 8.3 in http://arxiv.org/pdf/1105.2883.pdf

In a slightly different direction. Erbar and Huesmann have recently studied http://link.springer.com/article/10.1007/s00526-014-0790-1 the curvature of the "configuration space" of a manifold and shown that there, there is a natural choice of measure which inherits lower Ricci curvature bounds from the underlying manifold. The configuration space is quite similar to the space of all probability measures (for example, the set of finite sums of dirac measures is dense in the space of probability measures with respect to weak convergence).