# Preprint of the project: SFB 701: Spectral Structures and Topological Methods in Mathematics - Project B3

## 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 $$C_{\mathrm{b},\theta}(\mathbb{R}^d,\mathbb{C}^N)$$.