  Let $v$ be a vector field in a bounded open set $G\subset {\mathbb {R}}^d$.
Suppose that $v$ is observed with a random noise at random points $X_i,
i=1,...,n,$ that are independent and uniformly distributed in $G.$ The problem
is to estimate the integral curve of the differential equation
\[\frac{dx(t)}{dt}=v(x(t)),\qquad t\geq 0,x(0)=x_0\in G,\] starting at a given
point $x(0)=x_0\in G$ and to develop statistical tests for the hypothesis that
the integral curve reaches a specified set $\Gamma\subset G.$ We develop an
estimation procedure based on a Nadaraya--Watson type kernel regression
estimator, show the asymptotic normality of the estimated integral curve and
derive differential and integral equations for the mean and covariance function
of the limit Gaussian process. This provides a method of tracking not only the
integral curve, but also the covariance matrix of its estimate. We also study
the asymptotic distribution of the squared minimal distance from the integral
curve to a smooth enough surface $\Gamma\subset G$. Building upon this, we
develop testing procedures for the hypothesis that the integral curve reaches
$\Gamma$. The problems of this nature are of interest in diffusion tensor
imaging, a brain imaging technique based on measuring the diffusion tensor at
discrete locations in the cerebral white matter, where the diffusion of water
molecules is typically anisotropic. The diffusion tensor data is used to
estimate the dominant orientations of the diffusion and to track white matter
fibers from the initial location following these orientations. Our approach
brings more rigorous statistical tools to the analysis of this problem
providing, in particular, hypothesis testing procedures that might be useful in
the study of axonal connectivity of the white matter.
