SRQ seminar – November 16th, 2018

Notes from a talk in the SRQ series.

SRQ Seminar 20181116 Mitter (partial transcript)

Fixpoint argument

We have two recursions

\(\gamma < 1\), \(u_k = (\tilde{g}_k, \tilde{\mu}_k)\)

\(\displaystyle \left\{ \begin{array}{l} \tilde{g}_{k + 1} = \gamma \tilde{g}_k - \tilde{\xi} (u_k)\\ \mu_{k + 1} = L^{\alpha} \mu_k + \rho (u_k) \end{array} \right.\)

Solving these equations we have (solve \(\tilde{g}\) forward and \(\mu\) backward)

\(\displaystyle \left\{ \begin{array}{l} \tilde{g}_k = \gamma^k \tilde{g}_0 - \sum_{j < k} \gamma^{j - k} \tilde{\xi} (u_j)\\ \mu_k = L^{- \alpha (N - k)} \mu_N + \sum_{k \leqslant j < N} L^{- \alpha (j - k)} \rho (u_j) \end{array} \right.\)

We are solving with the condition that coupling constant are bounded. So we can take the limit \(N \mathop{\rightarrow}\limits \infty\) and obtain

\(\displaystyle \mu_k = \sum_{k \leqslant j < \infty} L^{- \alpha (j - k)} \rho (u_j) .\)

We consider now the whole trajectory \(v = (u_k, k \geqslant 0)\) and the two equations have the form

\(\displaystyle v = F_{\tilde{g}_0} (v)\)

where the \(\tilde{g}_0\) in the recursion is considered an external parameter.

So we get a fixpoint problem in the space of trajectories with norm \(\| v \| = \sup_k | u_k |\). Solution is \(v_{\ast}\) and at the end we obtain a map \(\tilde{g}_0 \rightarrow \mu_0 = \mu_0 (\tilde{g}_0)\).