Nonprobabilistic analogues of the Cauchy process

It is known that a solution of the Cauchy problem for an evolution equation having a convolution operator with a generalized function $|x|^{-2}$, in the right-hand side admits a probabilistic representation in the form of the expectation of a trajectory functional of the Cauchy process. We construct similar representations for evolution equations having a convolution operator with a generalized function $(-1)^m|x|^{-2m-2}$ for arbitrary $m\in\mathbf{N}$.