Paper VI: Reversible coarse dynamics

R2 Reversible coarse dynamics

On this page · Permalink · Source line 5347

Let \(Q\) be a finite Markov generator acting on column probability vectors, let \((B_a)\) be a partition into nonempty blocks, and let \(C_{ai}={{\boldsymbol{1}}}_{i\in B_a}\). Strong lumpability means \(CQ=\overline Q C\) for a coarse Markov generator. For observables, the backward generator is \(Q^T\); weighted \(L^2\) inner products use the stated stationary probability.

Permalink · Source line 5348

Theorem R2.1: Reversibility descends

Assume \(Q\) is strongly lumpable and reversible with respect to a positive stationary distribution \(\pi\), so

\[ Q_{ij}\pi_j=Q_{ji}\pi_i. \]

Define

\[ \overline\pi_a=\sum_{i\in B_a}\pi_i. \]

Then \(\overline\pi\) is stationary for \(\overline Q\), and

\[ \overline Q_{ab}\overline\pi_b = \overline Q_{ba}\overline\pi_a. \]

Thus the coarse chain is reversible.

Read proof · Read statement and proof side by side · Permalink · Source line 5349 · Claim record · Lean coverage · Code · Worked examples

Proof of Theorem R2.1

Stationarity follows from

\[ \overline Q\overline\pi =\overline QC\pi =CQ\pi=0. \]

For reversibility, use the aggregate rate formula:

\[ \begin{aligned} \overline Q_{ab}\overline\pi_b &=\sum_{j\in B_b}\pi_j\sum_{i\in B_a}Q_{ij}\\ &=\sum_{i\in B_a}\sum_{j\in B_b}\pi_iQ_{ji}\\ &=\overline Q_{ba}\overline\pi_a. \end{aligned} \]

Back to statement · Read statement and proof side by side · Permalink · Source line 5367 · Claim record · Lean coverage · Code · Worked examples

Corollary R2.2: Coarse Dirichlet form

Under the hypotheses of Theorem R2.1, let \(Jg\) be the fine function that is constant with value \(g_a\) on \(B_a\). Then

\[ -{\left\langle Jg,Q^TJg\right\rangle}_{L^2(\pi)} = -{\left\langle g,\overline Q^Tg\right\rangle}_{L^2(\overline\pi)}. \]

Read proof · Read statement and proof side by side · Permalink · Source line 5385 · Claim record · Lean coverage · Code · Worked examples

Proof of Corollary R2.2

Expand the left side and group the fine indices by their coarse blocks. The aggregate rate criterion converts the resulting sums into the coarse generator entries.

Back to statement · Read statement and proof side by side · Permalink · Source line 5394 · Claim record · Lean coverage · Code · Worked examples

Source and conversion record

Source SHA-256: 736e615131f57daf341b6cb0600761d58950450947c85d433206148cca228333

Unsupported prose commands stop the build. The mathematical expression sequence is checked against the source, with manuscript macros expanded for display.

Conversion records

Read and explore

Proofs and implementation

About this work

Other resources

R2 Reversible coarse dynamics

Theorem R2.1: Reversibility descends

Proof of Theorem R2.1

Corollary R2.2: Coarse Dirichlet form

Proof of Corollary R2.2