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Statistical exponential formulas for homogeneous diffusion

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  • Let $\Delta^{1}_{p}$ denote the $1$-homogeneous $p$-Laplacian, for $1 \leq p \leq \infty$. This paper proves that the unique bounded, continuous viscosity solution $u$ of the Cauchy problem \begin{eqnarray} u_{t} - ( \frac{p}{ N + p - 2 } ) \Delta^{1}_{p} u = 0 \quad \mbox{for} \quad x \in R^N \quad \mbox{and} \quad t > 0 , \\ \\ u(\cdot,0) = u_0 \in BUC(R^N). \end{eqnarray} is given by the exponential formula \begin{eqnarray} u(t) := \lim_{n \to \infty}{ ( M^{t/n}_{p} )^{n} u_{0} } \ , \end{eqnarray} where the statistical operator $M^h_p \colon BUC( R^{N} ) \to BUC( R^{N} )$ is defined by \begin{eqnarray} (M^{h}_{p} \varphi)(x) := (1-q) median_{\partial B(x,\sqrt{2h})}{ \{ \varphi \} } + q \int_{\partial B(x,\sqrt{2h})}{ \varphi ds } \end{eqnarray} when $1 \leq p \leq 2$, with $q := \frac{ N ( p - 1 ) }{ N + p - 2 }$, and by \begin{eqnarray} (M^{h}_{p} \varphi )(x) := ( 1 - q ) midrange_{\partial B(x,\sqrt{2h})}{ \{ \varphi\} } + q \int_{\partial B(x,\sqrt{2h})}{ \varphi ds } \end{eqnarray} when $p \geq 2$, with $q = \frac{ N }{ N + p - 2 }$. Possible extensions to problems with Dirichlet boundary conditions are mentioned briefly.
    Mathematics Subject Classification: Primary: 35K, 47H20; Secondary: 37L05, 47H05.

    Citation:

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