Solving a SDE / Finding expectation Value

Question

I am working on a physics problem, and have come across the following stochastic differential equation:

$dX(t) = \left( \frac{8}{3} X(t) - 3 X(t)^3\right)dt + dW$.

I have tried all the methods to solve this SDE (reduction, change of variables, etc...) and have even settled for evaluating the expectation $\mathbb{E}[X(t)]$, but the cubic term in the drift coefficient is proving to be very problematic!!

By the way, I am fully aware that global solutions may not exist/be unique, since the drift coefficient implies blow-up of solutions after some explosion time, I am curious strictly about local solutions.

Any thoughts?

Thanks!

Answer 1

This model is well know as "double-well" potential. The closed form solution to $\mathbb{E}(X(t))$ is not known. See Iacus, ch. 1.13.8.

Physically, you would expect that the global solution exists and is unique, since the drift term pushes the particle back to area around the origin every time $X(t)^2 > \tfrac{8}{9}$.

We can proceed with simulation, using Mathematica, using initial condition $X(0) = \frac{2}{3} \sqrt{2}$, using Roessler's 3-stage stochastic Runge-Kutta scheme with strong order of $\frac{3}{2}$ (see ch. 6.3 of doi:10.1137/09076636X):

The stationary distribution is symmetric with zero mean.

In fact, it is easy to find the pdf of the stationary distribution, see ch. 1.10.1 of Iacus. Using v10.2:

SDEStationaryDistribution[drift_, diff_, {x_, min_, max_}] := 
 Module[{s, y},
  s = Exp[-2 Integrate[drift/diff^2, {x, 0, y}, 
      Assumptions -> min < y < max]];
  ProbabilityDistribution[1/((s /. y -> x) diff^2), {x, min, max}, 
   Method -> "Normalize"]]

we have:

Hope this helps.