The Doss trick on symmetric spaces (Q2571655)

In the introduction the idea underlying the Doss trick is described as follows. Consider a Wiener process \((W_t)_{t\geq 0}\) on \(\mathbb{R}^n\) and its rotation \((\sqrt{i}W_t)_{t\geq 0}\) to \(\mathbb{C}^n\). Using the Feynman-Kac formula \[ \psi(t,x)= E(e^{-i\int^t_0 V(x+ \sqrt{i} W_{t-\tau})\,d\tau} \psi_0(+ \sqrt{i}W_{t-\tau}))\tag{1} \] \textit{H. Doss} [Commun. Math. Phys. 73, 247--264 (1980; Zbl 0427.60099)] demonstrated, under appropriate assumptions on \(V\) and the initial condition \(\psi_0\), that a solution to the Schrödinger equation can be given in terms of (1), at least for \(t\), \(x\) in a subset of \([0,\infty)\times\mathbb{R}^n\). In the paper of \textit{H. Thaler} [Potential Anal. 18, No. 2, 119--140 (2003; Zbl 1012.22016)] it has been shown that on compact connected semisimple Lie groups the same idea gives solutions of the Schrödinger equations by constructing a process on the complexification of the Lie group. Other important works are, for example, \textit{S. Albeverio, Z. Brzeźniak} and \textit{Z. Haba} [Potential Anal. 9, 65--82 (1998; Zbl 0928.58016)] and \textit{H. Doss} [Bull. Sci. Math. (2) 109, 179--208 (1985; Zbl 0564.35101)]. In this paper it is proved that the results obtained by \textit{H. Thaler} [loc. cit.] carry over to symmetric spaces of compact type. This results in the fact that any Riemannian globally symmetric space is a homogeneous space \(G/K\), with \(G\) a Lie group and \(K\) a closed subgroup. The idea is to find a Markov process \((g_t)_{t> 0}\) on \(G_{\mathbb{C}}\), the complexification of \(G\), possessing all the necessary properties, and to project it to a process \((\gamma_t)_{t\geq 0}\) on \(G_{\mathbb{C}}/K_{\mathbb{C}}\), the complexification of \(G/K\). The process \((\gamma_t)_{t\geq 0}\) enters the probabilistic representation of the solutions of the Schrödinger equation. In Section 2 ``Preliminaries'', the author presents some basic definitions and results on the complexification of Lie groups, and some isomorphisms of invariant function spaces are established. In Section 3, the process \((\gamma_t)_{t> 0}\) on \(G_{\mathbb{C}}/K_{\mathbb{C}}\) is constructed, which gives rise to a semigroup \((P_t)_{t\geq 0}\). In Section 4 ``Schrödinger equations with potential'', it is shown that for \(w\in\Omega\) the restriction of \((\overline P_t w)(\overline a)= \mathbb{E}(w(\gamma^{\overline a}_t))\), \(\overline a\in G_{\mathbb{C}}/K_{\mathbb{C}}\), to \(G/K\) solves the Schrödinger equation, i.e. \[ {d\over dt} (\overline P_t w)|_{G/K}= i\Delta_{G/K} (\overline P_t w)|_{G/K}. \] In Section 5 ``Perturbation by a potential'', the author considers the general case including potentials \(V\in\Omega\). The main result of this paper is given in Theorem 1, where it is stated that the Schrödinger equation is satisfied not only in the \(L^2\)-sense but also in a subspace of continuous functions. For other details see the author's references.