Stationary probability distribution for nonlinear dynamical systems with noise

Conclusion

As shown in the preceding sections, the method for calculating the stationary probability density is mathematically exact (with the exception of the use of the saddle-point method, which is an excellent approximation). In fact, the linear integral equation (37) is equivalent to the linear differential equation (22); because of the smoothing effect of integration, its solution must be much easier. This does not mean that it will be easy. We may expect serious difficulties in numerical calculations: sincez(x)=p ₀ (x)/q ₀(x), andq ₀(x) ∼ exp (−ux ²) is vanishingly small wherep ₀ is not,z must take on enormous values. For practical purposesu may be lessened to such a value thatu ^−1/2 is approximately equal to the accuracy with which we can localize points in the phase space, i.e., with which we can measure the velocity of the fluid. Even so, some sophisticated method of calculation will be needed. Another limitation is connected with the dimensionality of the equation; computational effort for solving integral equations grows exponentially with their dimension. Still, the method promises that an explicit calculation of the stationary probability distribution for some simple realistic nonlinear system may be performed.