Probabilistic representation of solutions to the parabolic $p$-Laplace equation
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866914516856471552 |
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| author | Barbu, Viorel Röckner, Michael |
| author_facet | Barbu, Viorel Röckner, Michael |
| contents | This work is concerned with the probabilistic representation of solutions to the $p$-Laplace evolution equation $\frac{\partial u}{\partial t}={\rm div}(|\nabla u|^{p-2}\nabla u)$ in $(0,\infty)\times\mathbb{R}^d$, $u(0,x)=u_0(x),$ $x\in\mathbb{R}^d$. One proves that, if $p\geq 4$, and if $u_0$ is a probability density with compact support and $u_0\in L^2$, $|\nabla u_0|\in L^\infty$, then $u$ can be represented as $u(t,x)dx=\mathscr L_{X(t)}(dx)$, where $\mathscr L_{X(t)}$ denotes the time marginal law of $X$ at time $t$ with $X$ being a probabilistically weak solution to a corresponding McKean-Vlasov stochastic differential equation. This result is based on a new second order global regularity result for the weak solutions to the parabolic $p$-Laplace equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26719 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Probabilistic representation of solutions to the parabolic $p$-Laplace equation Barbu, Viorel Röckner, Michael Analysis of PDEs Probability This work is concerned with the probabilistic representation of solutions to the $p$-Laplace evolution equation $\frac{\partial u}{\partial t}={\rm div}(|\nabla u|^{p-2}\nabla u)$ in $(0,\infty)\times\mathbb{R}^d$, $u(0,x)=u_0(x),$ $x\in\mathbb{R}^d$. One proves that, if $p\geq 4$, and if $u_0$ is a probability density with compact support and $u_0\in L^2$, $|\nabla u_0|\in L^\infty$, then $u$ can be represented as $u(t,x)dx=\mathscr L_{X(t)}(dx)$, where $\mathscr L_{X(t)}$ denotes the time marginal law of $X$ at time $t$ with $X$ being a probabilistically weak solution to a corresponding McKean-Vlasov stochastic differential equation. This result is based on a new second order global regularity result for the weak solutions to the parabolic $p$-Laplace equation. |
| title | Probabilistic representation of solutions to the parabolic $p$-Laplace equation |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2604.26719 |