The problem of finding a positive distribution, which corresponds to a given complex density in complex Langevin approach, is studied. By the requirement that the moments of the positive distribution and of the complex density are equal, one can reduce the problem to solving the matching conditions. After imposing the positivity of the distribution, these conditions become a set of quadratic equations. Groebner basis approach and minimization methods were used to find its approximate solutions, when the set of equations is restricted to a few lowest-order moments. The problem is ambiguous and the number of solutions rapidly increases with the number of equations. For a Gaussian complex density, these solutions are compared with the exact result, which is known in this special case.

Theoretical Developments