Gaussian estimates for general parabolic operators in dimension 1

TL;DR

This paper establishes Gaussian bounds for solutions of general one-dimensional parabolic equations with bounded measurable coefficients, using a canonical form and eigenfunction techniques.

Contribution

It introduces a method to derive Gaussian estimates for a broad class of parabolic operators in one dimension, including a novel use of a corrector function.

Findings

Derived Gaussian estimates for fundamental solutions.

Established Holder continuity for solutions of the canonical equation.

Connected solutions to a generalized eigenfunction to simplify analysis.

Abstract

We derive in this paper Gaussian estimates for a general parabolic equation $u_{t}-\big(a(x)u_{x}\big)_x= r(x)u$ over $\mathbb{R}$. Here $a$ and $r$ are only assumed to be bounded, measurable and $\mathrm{essinf}_{\mathbb{R}} a>0$. We first consider a canonical equation $\nu (x) \partial_{t}p - \partial_{x }\big( \nu (x)a(x)\partial_{x}p\big)+W\partial_{x}p=0$, with $W\in \mathbb{R}$, $\nu$ bounded and $\mathrm{essinf}_{\mathbb{R}} \nu>0$, for which we derive Gaussian estimates for the fundamental solution: $$\forall t>0, x,y\in \mathbb{R}, \quad \displaystyle\frac{1}{Ct^{1/2}}e^{-C|T(x)-T(y)-Wt|^{2}/t} \leq P(t,x,y)\leq \frac{C}{t^{1/2}}e^{-|T(x)-T(y)-Wt|^{2}/Ct}.$$ Here, the function $T$ is a corrector, for which we are able to derive appropriate properties using one-dimensional arguments. We then show that any solution $u$ of the original equation could be divided by some generalized principal eigenfunction $\phi_\gamma$ so that $p:=u/\phi_\gamma$ satisfies a canonical equation. As a byproduct of our proof, we derive Nash type estimates, that is, Holder continuity in $x$, for the solutions of the canonical equation.