Absolute continuity of heat kernel measure with pinned Wiener measure on loop groups
Let \(K\) be a compact Lie group, with unity \(e\) and Lie algebra \(k\), endowed with an Ad-invariant inner product. Let \(\beta(t,s)\) and \(\gamma(t,s)\) be, respectively, a \(k\)-valued Brownian sheet and a \(k\)-valued Brownian bridge sheet. Let \(g_s\) denote the solution to the Stratonovitch equation on \(K\): \(dg_s=g_s\beta(t,\delta s)\), with \(g_0=e\); so that \(g_s\) is a Brownian motion on \(K\) with variance \(t\). Denote by \(\gamma_t\) its law, that is to say the Wiener measure with variance \(t\). The pinned Wiener measure \(\gamma^0_t\) on \(K\) with variance \(t\) is deduced from \(\gamma_t\) by Doob \(h\)-transform with respect to the heat kernel \(p_t\) on \(K\). Of course \(\gamma_t\) is a law on the path group \(W(K)\) and \(\gamma^0_t\) a law on the loop group \({\mathcal L}(K)\). Consider then the two continuous processes \(S(t,s)\) and \(S^0(t,s)\) solving: \(S(\delta t, s)=S(t,s)\beta(\delta t,s)\); \(S^0(\delta t,s)=S^0(t,s)\gamma(\delta t,s)\); \(S(0,s)=S^0(0,s)=e\). Denote by \(\nu_t\) and \(\nu^0_t\) the laws of \(S(t,\cdot)\) and \(S^0(t,\cdot)\), respectively, which are the heat kernel measures with variance \(t\) on \(W(K)\) and \({\mathcal L}(K)\), respectively. Airault and Malliavin proved that \(\nu_t=\gamma_t\) on \(W(K)\). The analogous result cannot hold in general at the level of \(\nu^0_t\) and \(\gamma^0_t\), on \({\mathcal L}(K)\), but the main result here is that \(\nu^0_t\) is absolutely continuous with respect to \(\gamma^0_t\), with Radon-Nikodym derivative bounded by \((2\pi t)^{(\dim K)/2}\times \rho_t(e)\). The proof is based on a computation by Airault and Malliavin of the infinitesimal generator associated with \(\gamma^0_t\), and on the maximum principle. In the particular case \(K=\mathbb{S}^1\), the density \(d\nu^0_t/d\gamma^0_t\) is explicitly computed.
scientific article; zbMATH DE number 1905998
scientific article