Integration by parts and Bismut-Elworthy-Li-type formula on the Heisenberg Lie group using the hypoelliptic measure

Abstract We study left-invariant vector fields defined on the Heisenberg Lie group and the associated stochastic differential equation, expressed in terms of a hypoelliptic operator. Utilizing the Bismut tangent space, we derive a new integration by parts formula on the path space of the Heisenberg Lie group using the hypoelliptic measure. Additionally, we establish a Bismut-Elworthy-Li-type formula for the hypoelliptic case in the Heisenberg Lie group. Finally, we extend our findings by calculating a Bismut-Elworthy-Li-type formula on the n -dimensional Lie group and considering the hypoelliptic case for a more general structure. These results provide new insights into the geometric analysis of stochastic processes on Lie groups, with potential applications in areas such as mathematical finance, control theory, and stochastic physics.