Nonprobabilistic Analogs of the Cauchy Process

It is known that a solution to the Cauchy problem for the evolution equation, the right-hand side of which contains a convolution operator with generalized function |x|−2, admits a probabilistic representation in the form of the expectation of the trajectory functional of the Cauchy process. Similar representations are constructed for evolution equations containing convolution operator with generalized function (−1)m|x|−2m − 2 for arbitrary m ∈ N.