  We propose to take advantage of using the Wiener path integrals as the formal
solution for the joint probability densities of coupled Langevin equations
describing particles suspended in a fluid under the effect of viscous and
random forces. Our obtained formal solution, giving the expression for the
Lyapunov exponent, i) will provide the description of all the features and the
behaviour of such a system, e.g. the aggregation phenomenon recently studied in
the literature using appropriate approximations, ii) can be used to determine
the occurrence and the nature of the aggregation - non-aggregation phase
transition which we have shown for the one-dimensional case and iii) allows the
use of a variety of approximative methods appropriate for the physical
conditions of the problem such as instanton solutions in the WKB approximation
in the aggregation phase for the one-dimensional case as presented in this
paper. The use of instanton approximation gives the same result for the
Lyapunov exponent in the aggregation phase, previously obtained by other
authors using a different approximative method. The case of non-aggregation is
also considered in a certain approximation using the general path integral
expression for the one-dimensional case.
