Paper VI: Fractional modes and killed transfer

R4 Fractional modes and killed transfer

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For the one-dimensional Dirichlet difference operator, let \(h=1/M\) with an integer \(M\geq2\), let \(1\leq k\leq M-1\), and set \(\lambda_{k,h}=4h^{-2}\sin^2(k\pi h/2)\). The retained explicit coefficient below is valid for every discrete mode.

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Corollary R4.1: Fractional low mode dispersion

Let \(0<\alpha\leq1\). Then

\[ 0 \leq (k\pi)^{2\alpha}-\lambda_{k,h}^\alpha \leq \frac{\alpha(4k^2)^{\alpha-1}(k\pi)^4}{12}h^2. \]

For every \(t\geq0\),

\[ {\left\lvert {\mathrm e}^{-t\lambda_{k,h}^\alpha} - {\mathrm e}^{-t(k\pi)^{2\alpha}} \right\rvert} \leq \frac{t\alpha(4k^2)^{\alpha-1}(k\pi)^4}{12}h^2. \]

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Proof of Corollary R4.1

The chord bound \(\sin x\geq2x/\pi\) on \([0,\pi/2]\) gives \(\lambda_{k,h}\geq4k^2\). Since \(x\mapsto x^\alpha\) is increasing and has decreasing derivative,

\[ (k\pi)^{2\alpha}-\lambda_{k,h}^\alpha \leq \alpha\lambda_{k,h}^{\alpha-1} \bigl((k\pi)^2-\lambda_{k,h}\bigr). \]

Use Theorem E12.5 and the lower bound for \(\lambda_{k,h}\). The exponential estimate follows from

\[ {\left\lvert{\mathrm e}^{-ta}-{\mathrm e}^{-tb}\right\rvert} \leq t{\left\lvert a-b\right\rvert} \]

for nonnegative \(a\) and \(b\).

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Let \(Q\) and \(\overline Q\) be finite Markov generators in the column convention. All times in this section satisfy \(t\geq0\).

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Theorem R4.2: Potential perturbation bound

Let \(V\) and \(W\) be nonnegative diagonal matrices. In the supremum norm,

\[ \boxed{ {\left\lVert {\mathrm e}^{t(Q^T-V)} - {\mathrm e}^{t(Q^T-W)} \right\rVert}_{\infty\to\infty} \leq t{\left\lVert V-W\right\rVert}_{\infty\to\infty}. } \]

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Proof of Theorem R4.2

Both killed Markov semigroups are contractions in the supremum norm. Apply the Duhamel identity with generator defect \(W-V\).

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Theorem R4.3: Coarse Feynman Kac transfer

Assume \(CQ=\overline QC\), and let \(J=C^T\) lift a coarse function to the corresponding block constant fine function. Suppose the potential is block constant, so

\[ VJ=J\overline V. \]

Then

\[ \boxed{ {\mathrm e}^{t(Q^T-V)}J = J{\mathrm e}^{t(\overline Q^T-\overline V)} } \]

for every \(t\geq0\).

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Proof of Theorem R4.3

Transpose the Markov intertwining relation to obtain

\[ Q^TJ=J\overline Q^T. \]

Together with \(VJ=J\overline V\), this gives

\[ (Q^T-V)J=J(\overline Q^T-\overline V). \]

The exponential intertwining follows from the power series.

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