English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this work we study the degenerate diffusion equation$\partial_{t}=x^{\alpha}a\left(x\right)\partial_{x}^{2}+b\left(x\right)\partial_{x}$for $\left(x,t\right)\in\left(0,\infty\right)^{2}$, equipped with a Cauchyinitial data and the Dirichlet boundary condition at $0$. We assume that theorder of degeneracy at 0 of the diffusion operator is$\alpha\in\left(0,2\right)$, and both $a\left(x\right)$ and $b\left(x\right)$are only locally bounded. We adopt a combination of probabilistic approach andanalytic method: by analyzing the behaviors of the underlying diffusionprocess, we give an explicit construction to the fundamental solution$p\left(x,y,t\right)$ and prove several properties for $p\left(x,y,t\right)$;by conducting a localization procedure, we obtain an approximation for$p\left(x,y,t\right)$ for $x,y$ in a neighborhood of 0 and $t$ sufficientlysmall, where the error estimates only rely on the local bounds of$a\left(x\right)$ and $b\left(x\right)$ (and their derivatives). There is arich literature on such a degenerate diffusion in the case of $\alpha=1$. Ourwork extends part of the existing results to cases with more general order ofdegeneracy, both in the analysis context (e.g., heat kernel estimates onfundamental solutions) and in the probability view (e.g., wellposedness ofstochastic differential equations).

Fundamental Solution to 1D Degenerate Diffusion Equation with Locally Bounded Coefficients