Exponentially weighted resolvent estimates for complex Ornstein–Uhlenbeck systems

Abstract

In this paper, we study differential operators of the form

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

for matrices \({A,B \in \mathbb{C}^{N,N}}\), where the eigenvalues of A have positive real parts. The sum \({A\triangle v(x)+ {\langle} Sx,\nabla v(x) \rangle}\) 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

$$\left[T(t)v\right](x) =\int_{\mathbb{R}^d}H(x,\xi,t)v(\xi){\mathrm{d}} \xi,\,x \in \mathbb{R}^d,\,t > 0,$$

in exponentially weighted function spaces. This is used to derive resolvent estimates for \({\mathcal{L}_{\infty}}\) in exponentially weighted L p-spaces \({L^p_{\theta}({\mathbb{R}}^d,\mathbb{C}^N)}\), \({1\leqslant p < \infty}\), as well as in exponentially weighted C b -spaces \({C_{b, \theta}({\mathbb{R}}^d,\mathbb{C}^N)}\).

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Correspondence to Denny Otten.

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D. Otten was supported by CRC 701 ‘Spectral Structures and Topological Methods in Mathematics’.

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Otten, D. Exponentially weighted resolvent estimates for complex Ornstein–Uhlenbeck systems. J. Evol. Equ. 15, 753–799 (2015). https://doi.org/10.1007/s00028-015-0279-1

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Mathematics Subject Classification

  • 35K45 (35J47, 35K08, 35B65, 35B45, 35B40)

Keywords

  • Heat kernel matrix
  • Ornstein–Uhlenbeck semigroup
  • Exponentially weighted function spaces
  • Resolvent estimates