# Angle of analyticity of semigroup

Is there any known parabolic PDEs in the literature where the angle of analyticity of the associated semigroup is $$<\pi/2$$ ?

For example, the angle of heat semigroup in $$L^2$$ is exactly $$=\pi/2$$. I'm wondering if there is a known example where the angle is e.g., $$\pi/4$$ or other value $$< \pi/2$$.

• Yes, such examples are known. For some references, see for instance Secion 7.2.6 in "W. Arendt: Semigroups and Evolution Equations: Functional Calculus, Regularity and Kernel Estimates (2004)". Dec 28, 2019 at 21:00
• @JochenGlueck Thank you. The example given is about the realization of the semigroup in $L^p$. Do you know any example in Hilbert space, say $L^2$ for example? Dec 28, 2019 at 22:46
• Good question... Are you interested in real coefficients only, or also in PDEs with complex coefficients? Dec 29, 2019 at 8:56
• Yes I'm interested in real coefficients, but if necessary no problem with complex ones. Dec 29, 2019 at 9:34
• With complex coefficients it's quite easy: just consider the PDE $\dot u = e^{i\frac{\pi}{4}} \frac{\partial^2}{\partial x^2} u$ on, say, $L^2(\mathbb{R})$ (though one might argue whether this is really a "parabolic" PDE). I don't know an example with real coefficients at the moment, but I would suspect that such examples exist. Dec 29, 2019 at 14:22

Too long for a comment. Actually I do not think that it is written anywhere but these kind of counterexamples are usually provided by the Ornstein-Uhlenbeck semigroup, generated by $$\Delta+Bx \cdot \nabla$$, where $$B$$ is a matrix. Assuming that all eigenvalues of $$B$$ have negative real parts, then an invariant measure $$\mu$$ exists (and is given by a Gaussian density). It turns out that the angle of analiticity in $$L^p$$ of the invariant measure can be computed exactly and can be smaller than $$\pi/2$$, even for $$p=2$$. This can be found in a paper by Chill, Fasangova, Pallara and myself.
To obtain similar examples in unweighted spaces one has only (but patiently) to compensate the weight thus obtaining an operator with a linear drift and a quadratic potential. This works however only from dimension $$2$$ on; in the one dimensional case the operator is always self-adjont but still the angle of analitycity is different from $$\pi/2$$ in $$L^p$$, for $$p$$ different from $$2$$. There is also a paper by E. Priola and myself dealing with non-analytic Markov semigroups where one finds other examples.
There are examples but you need singularities or unbounded coefficients; the uniformly parabolic case with regular coefficients is indeed a perturbation of the laplacian. In 1D, if you perturb the harmonic oscillator $$D^2-x^2$$ by a linear drift $$bxD$$, the angle of analyticity depends on $$b$$.