Pointwise upper bound for the fundamental solution of fractional Fokker-Planck equation

TL;DR

This work addresses the pointwise upper bound for the fundamental solution of the linear kinetic fractional Fokker-Planck equation $\partial_t f+v\cdot\nabla_x f+|D_v|^{2s} f=0$ with $0<s<1$. It derives an explicit kernel representation in Fourier space, introducing the phase function $M(\xi,\eta)$ and obtaining $\mathcal{K}(1,x,v)=\iint e^{-M(\xi,\eta)-i\xi\cdot x-i\eta\cdot v}\,d\xi d\eta$, where $M(\xi,\eta)=\int_{0}^{1}|(\xi-\eta)\tau+\eta|^{2s}\,d\tau=\frac{1}{2s+1}\frac{|\xi|^{2s}\xi-|\eta|^{2s}\eta}{\xi-\eta}$. With Littlewood-Paley decomposition, the authors decompose the kernel into dyadic pieces and derive sharp bounds for the frequency-localized contributions, handling the regimes $0<s\le\tfrac12$ and $\tfrac12<s<1$ separately. The main result provides a pointwise bound for derivatives of the fundamental solution at $t=1$, $|\partial_x^{b_1}\partial_v^{b_2}K(1,x,v)|\lesssim \langle x, x+v\rangle^{-(2+2s-2\varepsilon)}\langle x\rangle^{-\varepsilon-b_1}\langle x+v\rangle^{-\varepsilon-b_2}$ for any $\varepsilon>0$, and, by scaling, the corresponding bound for general times $t$. This LP-based approach yields a concise, self-contained route to well-posedness/regularity insights for fractional kinetic equations, distinct from stochastic-method proofs in related work. The results enhance understanding of the fundamental solution's decay properties in kinetic, fractional settings and offer tools for further analysis of related Kolmogorov–Fokker-Planck-type problems.

Abstract

In this paper, we investigate the fundamental solution of the fractional Fokker-Planck equation. Utilizing the Littlewood-Paley decomposition technology, we present a concise proof of the pointwise estimate for the fundamental solution.

Theorems & Definitions (11)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• proof : Proof of Theorem \ref{['estimate K']}