Hypoelliptic non-homogeneous diffusions
Let \[ \begin{cases} dx_t &= \sum^n_{j=1} X_j(t, x_t) dw^j_t+ X_0(t, x_t) dt,\quad t> s,\\ x_s &= x,\end{cases}\tag{1} \] be a stochastic differential equation in Stratonovich form, where \((w^1,\dots, w^m)\) is a standard Brownian motion and \(t\mapsto X_j(t,\bullet)\), \(j= 0,\dots, m\), are measurable flows of \({\mathcal C}^\infty\) vector fields on \(\mathbb{R}^d\). \(T_{st}\) denotes a non-homogeneous Markov semigroup \(T_{st}: f\mapsto E(f(x^s_t(x)))\), where \(x^s_t(x)\) is a solution of (1), with generator \(L_t={1\over 2} \sum^m_{j=1} X^2_j(t,\bullet)+ X_0(t,\bullet)\). The authors study the problem of the existence of densities for the laws \(\mu_{st}(x, dy)\) of solutions \(x^s_t(x)\). The parabolic-hypoellipticity of the operator \({\partial\over\partial t}+ L_t+ c\) is also proved.
scientific article; zbMATH DE number 1837003