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

The following stochastic Cauchy initial-value problem is studied for theparabolic heat equation on a domain $ \mathbf{Q}\subset{\mathbf{R}}^{n}$ withrandom field initial data. \begin{align} &{\square}\widehat{u(x,t)} \equiv\bigg(\frac{\partial}{\partialt}-{\Delta}_{x}\bigg)\widehat{u(x,t)}=0,~x\in\mathbf{Q},t> 0 \end{align}\begin{align} \widehat{u(x,0)}=\phi(x)+\mathscr{J}(x),~x\in\mathbf{Q},t=0\end{align} where $\phi(x)\in C^{\infty}({\mathbf{Q}})$, and $\mathscr{J}(x)$is a classical Gaussian random scalar field with expectation$\mathbb{E}[\![\mathscr{J}(x)]\!]=0 $ and with a regulated covariance$\mathbb{E}[\![ \mathscr{J}(x)\otimes\mathscr{J}(y)]\!]=\zeta J(x,y;\ell)$,correlation length $\ell$ and $\mathbb{E}[\![\mathscr{J}(x)\otimes\mathscr{J}(x)]\!]=\zeta<\infty$. The randomly perturbedsolution $\widehat{u(x,t)}$ is a stochastic convolution integral. This leads tostochastic extensions and versions of some classical results for the heatequation; in particular, a Li-Yau differential Harnack inequality \begin{align}\mathbb{E}\left[\!\!\left[\frac{|\nabla\widehat{u(x,t)}|^{2}}{|\widehat{u(x,t)}|^{2}}\right]\!\!\right]-\mathbb{E}\left[\!\!\left[\frac{\tfrac{\partial}{\partialt}\widehat{u(x,t)}}{\widehat{u(x,t)}} \right]\!\!\right]\le\frac{1}{2}n\frac{1}{t} \end{align} and a parabolic Harnack inequality. Decayestimates and bounds for the volatility$\mathbb{E}[\![|\widehat{u(x,t)}|^{2}]\!]$ and p-moments$\mathbb{E}[\![|\widehat{u(x,t)}|^{p}]\!] $ are derived. Since $\lim_{t\uparrow\infty}\mathbb{E}[\![|\widehat{u(x,t)}|^{p}]\!]=0 $, the Cauchy evolution ofthe randomly perturbed solution is stable since the heat equation smooths outor dissipates volatility induced by initial data randomness as$t\rightarrow\infty$.

Stochastic Cauchy Initial Value Formulation Of The Heat Equation For Random Field Initial Data: Smoothing, Harnack-Type Bounds And p-Moments