Numerical Analysis of equivariant evolution equations
14-052 Denny Otten.
Exponentially weighted resolvent estimates for complex Ornstein-Uhlenbeck systems
In this paper we study differential operators of the form
\[
\left[\mathcal{L}_{\infty}v\right](x) = A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle - Bv(x),\,x\in\mathbb{R}^d,\,d\geq 2,
\]
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
\[
\left[T(t)v\right](x) = \int_{\mathbb{R}^d}H(x,\xi,t)v(\xi)d\xi,\,x\in\mathbb{R}^d,\,t>0,
\]
in exponentially weighted function spaces. This is used to derive resolvent estimates for \(\mathcal{L}_{\infty}\) in exponentially weighted \(L^p\)-spaces \(L^p_{\theta}(\mathbb{R}^d,\mathbb{C}^N)\), \(1\leq p\lt \infty\), as well as in exponentially weighted \(C_{\mathrm{b}}\)-spaces \(C_{\mathrm{b},\theta}(\mathbb{R}^d,\mathbb{C}^N)\).
