The identification problem for complex-valued Ornstein–Uhlenbeck operators in $L^p(\mathbb {R}^d,\mathbb {C}^N)$

Abstract

In this paper we study perturbed Ornstein–Uhlenbeck operators

$$\begin{aligned} \left[ \mathcal {L}_{\infty } v\right] (x)=A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle -B v(x),\,x\in \mathbb {R}^d,\,d\geqslant 2, \end{aligned}$$

for simultaneously diagonalizable matrices $A,B\in \mathbb {C}^{N,N}$. The unbounded drift term is defined by a skew-symmetric matrix $S\in \mathbb {R}^{d,d}$. Differential operators of this form appear when investigating rotating waves in time-dependent reaction diffusion systems. We prove under certain conditions that the maximal domain $\mathcal {D}(A_p)$ of the generator $A_p$ belonging to the Ornstein–Uhlenbeck semigroup coincides with the domain of $\mathcal {L}_{\infty }$ in $L^p(\mathbb {R}^d,\mathbb {C}^N)$ given by

$$\begin{aligned} \mathcal {D}^p_{\mathrm {loc}}(\mathcal {L}_0)=\left\{ v\in W^{2,p}_{\mathrm {loc}}\cap L^p\mid A\triangle v + \left\langle S\cdot ,\nabla v\right\rangle \in L^p\right\} ,\,1<p<\infty . \end{aligned}$$

One key assumption is a new $L^p$-dissipativity condition

$$\begin{aligned} |z|^2\mathrm {Re}\,\left\langle w,Aw\right\rangle + (p-2)\mathrm {Re}\,\left\langle w,z\right\rangle \mathrm {Re}\,\left\langle z,Aw\right\rangle \geqslant \gamma _A |z|^2|w|^2\;\forall \,z,w\in \mathbb {C}^N \end{aligned}$$

for some $\gamma _A>0$. The proof utilizes the following ingredients. First we show the closedness of $\mathcal {L}_{\infty }$ in $L^p$ and derive $L^p$-resolvent estimates for $\mathcal {L}_{\infty }$. Then we prove that the Schwartz space is a core of $A_p$ and apply an $L^p$-solvability result of the resolvent equation for $A_p$. In addition, we derive $W^{1,p}$-resolvent estimates. Our results may be considered as extensions of earlier works by Metafune, Pallara and Vespri to the vector-valued complex case.