Exponentially weighted resolvent estimates for complex Ornstein-Uhlenbeck systems

TL;DR

This paper develops exponential resolvent estimates for complex Ornstein-Uhlenbeck operators, providing a foundation for analyzing the stability and decay of rotating wave solutions in reaction-diffusion systems.

Contribution

The paper constructs a heat kernel and derives resolvent estimates for complex Ornstein-Uhlenbeck operators in weighted spaces, extending previous results to complex matrices and unbounded drift terms.

Findings

Constructed a heat kernel matrix for the operator

Derived resolvent estimates in weighted $L^p$-spaces

Established exponential decay properties of solutions

Abstract

In this paper we study differential operators of the form \begin{align*} \left[\mathcal{L}_\infty v \right](x) = A\triangle v(x) + \left\langle Sx,\nabla v(x) \right\rangle - Bv(x), \,x \in \mathbb{R}^d, \,d \geqslant 2, \end{align*} for matrices $A,B\in\mathbb{C}^{N,N}$, where the eigenvalues of $A$ have positive real parts. The sum $A\triangle v(x) + \left\langle Sx, \nabla v(x) \right\rangle$ is known as the Ornstein-Uhlenbeck operator with an unbounded drift term defined by a skew-symmetric matrix $S\in\mathbb{R}^{d,d}$. Differential operators such as $\mathcal{L}_{\infty}$ arise as linearizations at rotating waves in time-dependent reaction diffusion systems. The results of this paper serve as foundation for proving exponential decay of such waves. Under the assumption that $A$ and $B$ can be diagonalized simultaneously we construct a heat kernel matrix $H(x,\xi,t)$ of $\mathcal{L}_{\infty}$ that solves the evolution equation $v_t=\mathcal{L}_{\infty}v$. In the following we study the Ornstein-Uhlenbeck semigroup \begin{align*} \left[ T(t)v\right](x) = \int_{\mathbb{R}^d} H(x,\xi,t) v(\xi) d\xi,\,x \in \mathbb{R}^d,\, t>0, \end{align*} in exponentially weighted function spaces. This is used to derive resolvent estimates for $\mathcal{L}_{\infty}$ in exponentially weighted $L^p$-spaces $L^p_{\theta} (\mathbb{R}^d,\mathbb{C}^N)$, $1\leqslant p<\infty$, as well as in exponentially weighted $C_{\mathrm{b}}$-spaces $C_{\mathrm{b},\theta}(\mathbb{R}^d,\mathbb{C}^N)$.