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We prove that if traceability conditions are fulfilled then a weak solution$h\in L^\infty(\R^+\times\R^d\times \R)$ to {the ultra-parabolic transportequation} \begin{equation*} \pa_t h + \Div_x\left(F(t,x,\lambda)h\right)=\sum\limits_{i,j=1}^k\pa^2_{x_ix_j}\left(b_{ij}(t,x,\lambda) h\right)+\pa_\lambda \gamma(t,x,\lambda),\end{equation*} is such that for every $\rho\in C^1_c(\R)$, the velocityaveraged quantity $\int_{\R}h(t,x,\lambda)$ $\rho(\lambda)d\lambda$ admits thestrong $L^1_{\rm loc}(\R^d)$-limit as $t\to 0$, i.e. there exist$h_0(x,\lambda)\in L^1_{\rm loc}(\R^d\times \R)$ and the set $E\subset\R^+$ offull measure such that for every $\rho\in C^1_c(\R)$, $$ L^1_{\rmloc}(\R^d)-\lim\limits_{t\to 0, \; t\in E} \int_{\R}h(t,x,\lambda)\rho(\lambda)d\lambda= \int_{\R} h_0(x,\lambda)\rho(\lambda)d\lambda. $$ As a corollary, under the traceability conditions, weprove existence of strong traces for entropy solutions to ultraparabolicequations in heterogeneous media.

Strong traces for averaged solutions of heterogeneous ultra-parabolic transport equations