A computable evolution equation for the joint response-excitation probability density function of stochastic dynamical systems

By using functional integral methods, we determine a computable evolution equation for the joint response-excitation probability density function of a stochastic dynamical system driven by coloured noise. This equation can be represented in terms of a superimposition of differential constraints, i.e. partial differential equations involving unusual limit partial derivatives, the first one of which was originally proposed by Sapsis & Athanassoulis. A connection with the classical response approach is established in the general case of random noise with arbitrary correlation time, yielding a fully consistent new theory for non-Markovian systems. We also address the question of computability of the joint response-excitation probability density function as a solution to a boundary value problem involving only one differential constraint. By means of a simple analytical example, it is shown that, in general, such a problem is undetermined, in the sense that it admits an infinite number of solutions. This issue can be overcome by completing the system with additional relations yielding a closure problem, which is similar to the one arising in the standard response theory. Numerical verification of the equations for the joint response-excitation density is obtained for a tumour cell growth model under immune response.