Stochastic intrinsic gradient flows on the Wasserstein space
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911627344871424 |
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| author | Ren, Panpan Röckner, Michael Wang, Feng-Yu Wittmann, Simon |
| author_facet | Ren, Panpan Röckner, Michael Wang, Feng-Yu Wittmann, Simon |
| contents | We construct stochastic gradient flows on the $2$-Wasserstein space $\mathcal P_2$ over $\mathbb R^d$ for energy functionals of the type $W_F(ρd x)=\int_{\mathbb R^d}F(x,ρ(x))d x$. The functions $F$ and $\partial_2 F$ are assumed to be locally Lipschitz on $\mathbb R^d\times (0,\infty)$. This includes the relevant examples of $W_F$ as the entropy functional or more generally the Lyapunov function of generalized porous media equations. First we define a class of Gaussian-based measures $Λ$ on $\mathcal P_2$ together with a corresponding class of symmetric Markov processes ${(R_t)}_{t\geq 0}$. Next, using Dirichlet form techniques we perform stochastic quantization for the perturbations of these objects which result from multiplying such a measure $Λ$ by a density proportional to $e^{-W_F}$. Finally we show that the intrinsic gradient $DW_F(μ)$ is defined for $Λ$-a.e. $μ$ and that the Gaussian-based reference measure $Λ$ can be chosen in such way that the distorted process ${(μ_t)}_{t\geq 0}$ is a martingale solution for the equation $dμ_t=-DW_F(μ_t) d t+d R_t$, $t\geq 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12755 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stochastic intrinsic gradient flows on the Wasserstein space Ren, Panpan Röckner, Michael Wang, Feng-Yu Wittmann, Simon Probability 60J60, 60J25, 60J46, 35Q84, 76S05 We construct stochastic gradient flows on the $2$-Wasserstein space $\mathcal P_2$ over $\mathbb R^d$ for energy functionals of the type $W_F(ρd x)=\int_{\mathbb R^d}F(x,ρ(x))d x$. The functions $F$ and $\partial_2 F$ are assumed to be locally Lipschitz on $\mathbb R^d\times (0,\infty)$. This includes the relevant examples of $W_F$ as the entropy functional or more generally the Lyapunov function of generalized porous media equations. First we define a class of Gaussian-based measures $Λ$ on $\mathcal P_2$ together with a corresponding class of symmetric Markov processes ${(R_t)}_{t\geq 0}$. Next, using Dirichlet form techniques we perform stochastic quantization for the perturbations of these objects which result from multiplying such a measure $Λ$ by a density proportional to $e^{-W_F}$. Finally we show that the intrinsic gradient $DW_F(μ)$ is defined for $Λ$-a.e. $μ$ and that the Gaussian-based reference measure $Λ$ can be chosen in such way that the distorted process ${(μ_t)}_{t\geq 0}$ is a martingale solution for the equation $dμ_t=-DW_F(μ_t) d t+d R_t$, $t\geq 0$. |
| title | Stochastic intrinsic gradient flows on the Wasserstein space |
| topic | Probability 60J60, 60J25, 60J46, 35Q84, 76S05 |
| url | https://arxiv.org/abs/2506.12755 |