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Let $\mathcal P_T:=\{\mu=(\mu_t)_{0\le t\le t}: \mu_t\in\mathcal P,~ \forall 0\le t\le 1\}$, where $\mathcal P$ is the space of probability measures on $\mathbb R$. Denote by $\rho$ the metric that is consistent to the weak convergence on $\mathcal P$, and define a distance $\rho_T$ on $\mathcal P_T$ by

$$\rho(\mu,\nu):=\sup_{0\le t\le 1} \rho(\mu_t,\nu_t),\quad \forall \mu,\nu\in\mathcal P_T.$$

My concern is to prove the existence of fixed point for some functional $F:\mathcal P_T\to\mathcal P_T$, where $F$ is assumed to be as good as possible, e.g. $F$ is continuous and maps some compact subset to itself. Does it exist the corresponding fixed-point theorem that can be applied to $\mathcal P_T$ (which is Polish, convex but NOT a vector space)?

Any answers, comments and references are appreciated!

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  • $\begingroup$ The wikipedia page on Schauder fixed-point theorem says: " In 1934, Tychonoff proved the theorem for the case when K is a compact convex subset of a locally convex space. This version is known as the Schauder–Tychonoff fixed-point theorem. B. V. Singbal proved the theorem for the more general case where K may be non-compact; the proof can be found in the appendix of Bonsall's book (see references)." en.wikipedia.org/wiki/Schauder_fixed-point_theorem $\endgroup$
    – Lars
    Commented Jul 19, 2021 at 23:02
  • $\begingroup$ @LarsvanderLaan Thank you very much for letting me know the reference $\endgroup$
    – GJC20
    Commented Jul 20, 2021 at 6:13

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