We consider a spatial domain
(d = 2 or 3), a time interval
a diffusion equation of the following form:
with initial conditions
denotes the space- and time-dependent diffusive field and
are describes external forces.
denotes the first time derivative and
the spatial gradient operator.
Furthermore, the scalar parameter
and the symmetric second-order diffusion tensor
are space-dependent coefficients.
can be related to a Wiener process using the relation
which direction-dependent smoothing lengths
For the special case of
corresponds to the standard deviation of the Gaussian smoothing in meters.
In the isotropic case,
simplifies to a scalar value, in which case we may re-write the diffusion equation as
and the isotropic smoothing length