Doob's ω-transform of parabolic problem for fractional Laplacian

Abstract

This paper studies the existence of nonnegative solutions for the following parabolic problem $\{\begin{array}{ll}{L}^{\omega }u+{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}}=0,& \text{in}{\mathbb{R}}^d\times (0,T),\\ u(x,0)=u_0(x),& \text{in}{\mathbb{R}}^d,\end{array}$where T>0, $L^{\omega }$ is a selfadjoint operator associated with a regular Dirichlet form Q, the initial value $u_0\in L^2({\mathbb{R}}^d)$, $u_0\ge 0$ is a Borel measurable function.

Keywords:

• Fractional Laplacian
• heat equation
• parabolic problem

Mathematics Subject Classifications:

• 35J35
• 35J50
• 35J60

Acknowledgements

We thank the referees for their times and comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).