论文标题
Hörmander的Markovian Rough Paths的版本
A Version of Hörmander's Theorem for Markovian Rough Paths
论文作者
论文摘要
我们考虑表格\（dy_t = \ sum_i v_i（y_t）d \ boldsymbol {x}^i_t+v_0（y_t）dt \）的粗略微分方程，其中\（\ boldsymbol {x} _t \）是一个巨大的粗糙路径。我们证明，如果矢量字段\（（（v_i）_ {0 \ leq i \ leq d} \）满足Hörmander的支架生成条件，则\（y_t \）承认具有高斯类型上限的平滑密度，鉴于\（x_t \）的生成器满足某些不合格的条件。本文的主要新成分是研究\（x_t \）的Jacobian过程的非分类性质。
We consider a rough differential equation of the form \(dY_t=\sum_i V_i(Y_t)d\boldsymbol{X}^i_t+V_0(Y_t)dt \), where \(\boldsymbol{X}_t \) is a Markovian rough path. We demonstrate that if the vector fields \((V_i)_{0\leq i\leq d} \) satisfy Hörmander's bracket generating condition, then \(Y_t\) admits a smooth density with a Gaussian type upper bound, given that the generator of \(X_t\) satisfy certain non-degenerate conditions. The main new ingredient of this paper is the study of non-degenerate property of the Jacobian process of \(X_t\).
