Statistical exponential formulas for homogeneous diffusion (1403.1853v1)

Abstract: 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 [ \left{ \begin{array}{c} u_{t} \ - \ ( \frac{p}{ \, N + p - 2 \, } ) \, \Delta^{1}_{p} u ~ = ~ 0 \quad \mbox{for} \quad x \in \mathbb{R}^{N}, \quad t > 0 \ \ u(\cdot,0) ~ = ~ u_{0} \in BUC( \mathbb{R}^{N} ) \end{array} \right. ] is given by the exponential formula [ u(t) ~ := ~ \lim_{n \to \infty}{ \left( M^{t/n}_{p} \right)^{n} u_{0} } \, ] where the statistical operator $M^{h}_{p} \colon BUC( \mathbb{R}^{N} ) \to BUC( \mathbb{R}^{N} )$ is defined by [ \left(M^{h}_{p} \varphi \right)(x) := (1-q) \operatorname{median}{\partial B(x,\sqrt{2h})}{ \left{ \, \varphi \, \right} } + q \operatorname{mean}{\partial B(x,\sqrt{2h})}{ \left{ \, \varphi \, \right} } \, ] with $q := \frac{ N ( p - 1 ) }{ N + p - 2 }$, when $1 \leq p \leq 2$ and by [ \left(M^{h}_{p} \varphi \right)(x) := ( 1 - q ) \operatorname{midrange}{\partial B(x,\sqrt{2h})}{ \left{ \, \varphi \, \right} } + q \operatorname{mean}{\partial B(x,\sqrt{2h})}{ \left{ \, \varphi \, \right} } \, ] with $q = \frac{ N }{ N + p - 2 }$, when $p \geq 2$. Possible extensions to problems with Dirichlet boundary conditions and to homogeneous diffusion on metric measure spaces are mentioned briefly.