An application of Itô's formula to an SDE on a Lie group

Question

I'm trying to understand a calculation in this paper (equation (3.8)). With some details removed, the setup is as follows.

Let $G$ be a Lie group, and $g(t)$ a curve in $G$ satisfying the SDE

$$dg(t) = T_e L_{g(t)}H_1 dW_t + T_e L_{g(t)}H_2 dt + T_e L_{g(t)}u(t) dt,$$ where $H_1,H_2$ are elements of the Lie algebra, $u(t)$ is a curve in the Lie algebra, and $W_t$ is a real-valued brownian motion. The SDE / stochastic integrals are in the Stratonovich sense. My knowledge of processes and SDE's on manifolds are from Elton Hsu's book. According to the construction in this book, a solution to an equation such as the above is defined by being a solution to the following euclidean SDE, for every smooth real-valued function f on G,

$$df(g(t)) = (T_e L_{g(t)}H_1)f\hspace{1mm} dW_t + (T_e L_{g(t)}H_2)f\hspace{1mm}dt + (T_e L_{g(t)}u(t))f \hspace{1mm} dt.$$

In the paper, they then consider a sort of perturbation of the curve g; $g(t)e_{\epsilon}(t)$, where e_{\epsilon}(t) is another curve in G. I don't think it is important to my question, but $e_\epsilon (t)$ is defined by a set of ODE's, equation (3.2) in the paper.

They then state that 'by Itö's formula' $g_\epsilon(t) := g(t) e_\epsilon(t)$ fulfills the SDE

$$dg_\epsilon(t) = T_e L_{g_\epsilon(t)}Ad_{e_{\epsilon}^{-1}(t)}(H_1) dW_t + T_e L_{g_\epsilon(t)}Ad_{e_{\epsilon}^{-1}(t)}H_2 dt + T_e L_{g_\epsilon(t)}Ad_{e_{\epsilon}^{-1}(t)}u(t) dt + T_e L_{g_\epsilon(t)} (T_{e_\epsilon (t)} L_{e^{-1}_\epsilon(t)}\dot{e}_\epsilon(t) dt $$

which (my calculation) is equivalent to

$$d(g(t) e_\epsilon (t)) = T_e (L_{g(t)}\circ R_{e_\epsilon (t)})H_1\hspace{1mm} dW_t + T_e (L_{g(t)}\circ R_{e_\epsilon (t)})H_2\hspace{1mm}dt + T_e (L_{g(t)}\circ R_{e_\epsilon (t)})u(t) \hspace{1mm} dt + T_{e_\epsilon (t)} L_{g(t)} \dot{e}_\epsilon(t) dt\hspace{1mm}$$

I haven't been able to verify this, or find hints elsewhere. I have a feeling that I'm missing (or lack knowledge of) something very simple. Any hints would be greatly appreciated. Even if only for the deterministic case ($H_1 = 0$).

Answer 1

To understand (3.8), I think it helps to write down (2.4) and (3.2) from that paper. From (3.2), note that $e_{\epsilon}(t)$ satisfies the ODE $$ \dot e_{\epsilon}(t) = \epsilon T L_{e_{\epsilon}(t)} \dot v(t) \;. \tag{1} $$ By the Euclidean Itô-Stratonovich conversion formula applied in the Lie algebra, note from (2.4) that $g(t)$ satisfies the Itô SDE $$ d g(t) = T L_{g(t)} \left( \sum_i H_i dW^i_t + u(t) dt \right) \;. \tag{2} $$

Note also that the product rule for Itô processes applied to $g_{\epsilon}(t) = g(t) e_{\epsilon}(t)$ reduces to the standard product rule because $e_{\epsilon}(t)$ has finite variation. Hence, from (1), (2), and the product rule, \begin{align} d g_{\epsilon}(t) &= T R_{e_{\epsilon}(t)} d g(t) + T L_{g(t)} \dot e_{\epsilon}(t) \\ &= T L_{g_{\epsilon}(t)} \left( \sum_{i} \operatorname{Ad}_{e_{\epsilon}(t)^{-1}} H_i dW^i_t + \operatorname{Ad}_{e_{\epsilon}(t)^{-1}} u(t) dt \right) + \epsilon T L_{g_{\epsilon}(t)} \dot v \tag{3} \end{align} where we used the identities \begin{align} T R_{e_{\epsilon}(t)} \; T L_{g(t)} = T L_{g(t)} \; T R_{\epsilon(t)} = T L_{g_{\epsilon}(t)} \; \operatorname{Ad}_{e_{\epsilon}(t)^{-1}} \;, \qquad T L_{g(t)} \dot e_{\epsilon}(t) = \epsilon T L_{g_{\epsilon}(t)} \dot v(t) \;. \end{align} Up to an Itô-Stratonovich correction term, (3) is exactly (3.8) in that paper.