Statistical exponential formulas for homogeneous diffusion

Abstract

Let ${\Delta }_p^1$ denote the $1$-homogeneous $p$-Laplacian, for $1\le p\le \infty$. This paper proves that the unique bounded, continuous viscosity solution $u$ of the Cauchy problem \[ \left\{ \begin{array}{c} u_{t} \ - \ ( \frac{p}{ \, N + p - 2 \, } ) \, Δ^{1}_{p} u ~ = ~ 0 \quad \mbox{for} \quad x \in \mathbb{R}^{N}, \quad t > 0 \\ \\ u(\cdot,0) ~ = ~ u_{0} \in BUC( \mathbb{R}^{N} ) \end{array} \right. \] 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 $M_p^h:BUC({\mathbb{R}}^N)\to BUC({\mathbb{R}}^N)$ is defined by \[ \left(M^{h}_{p} φ\right)(x) := (1-q) \operatorname{median}_{\partial B(x,\sqrt{2h})}{ \left\{ \, φ\, \right\} } + q \operatorname{mean}_{\partial B(x,\sqrt{2h})}{ \left\{ \, φ\, \right\} } \, \] with $q:=\frac{N(p-1)}{N+p-2}$, when $1\le p\le 2$ and by \[ \left(M^{h}_{p} φ\right)(x) := ( 1 - q ) \operatorname{midrange}_{\partial B(x,\sqrt{2h})}{ \left\{ \, φ\, \right\} } + q \operatorname{mean}_{\partial B(x,\sqrt{2h})}{ \left\{ \, φ\, \right\} } \, \] with $q=\frac{N}{N+p-2}$, when $p\ge 2$. Possible extensions to problems with Dirichlet boundary conditions and to homogeneous diffusion on metric measure spaces are mentioned briefly.

References in corpus (1)

• Asymptotic statistical characterizations of p-harmonic functions of two variables