Hypoelliptic heat kernels on infinite-dimensional Heisenberg groups (Q2796513)

Consider an abstract Wiener space \((W,H,\mu)\), a finite dimensional real Hilbert space \(C\), and a continuous skew-symmetric bilinear form \(\omega:W\times W\to C\). Then a Banach Lie group structure can be obtained on \(G=W\times C\) by considering the multiplication NEWLINE\[NEWLINE(\omega_1,c_1)(\omega_2,c_2)=(\omega_1+\omega_2,c_1+c_2+\frac12\omega(\omega_1,\omega_2)).NEWLINE\]NEWLINE This group is called Heisenberg-like by the authors, and \(H\times C\) is the Cameron-Martin subgroup. Then one is able to deduce a notion of hypoelliptic Brownian motion NEWLINE\[NEWLINEg_t=(B_t,\frac12\int_0^t\omega(B_s,dB_s))NEWLINE\]NEWLINE on \(G\), extending the Brownian motion and Lévy area of the finite dimensional case.NEWLINENEWLINEThe first main result of this work is the formula NEWLINE\[NEWLINE\nu_T(dx,dc)=\gamma_T(x,c)\mu_T(dx)m(dc)NEWLINE\]NEWLINE where \(\nu_T\) is the law of \(g_T\), \(\mu_T\) is the dilated Wiener measure \(\mu_T(A)=\mu(T^{-1}A)\), \(m\) is the Lebesgue measure on \(C\), and \(\gamma_T\) is given as an explicit conditional expectation.NEWLINENEWLINEThen, the authors deduce Fernique-type integrability results about \(g_T\), prove that the heat kernel is quasi-invariant under translations by the Cameron-Martin subgroup, and is infinitely differentiable in directions of this subgroup.