Finding Exit Times for an Into Stochastic Process

Finding Exit Times for an Into Stochastic Process

The following is the Ito stochastic Process that I am interested in (where
z(t)=[x(t),y(t)]:

with:
b1b = 0.9;
b3b = .8;
a1b = 0.1;
a3b = 0.2;
eps = 0.1;
G = (1/eps^2)*b1b ; a1 = (1/eps^2)*a1b; a3 = (1/eps^2)*a3b;
xc = Sqrt[a1/a3];
Uc = a1*xc^2/2 - a3*xc^4/4
To find the first exit time $\tau$ maybe the following can be used
(courtesy of b.gatessucks) -- it is a Mathematica code:
CONSTRAINT FOR EXIT TIME:

const[x_, y_] := And[10^-8 <= y <= 10^-3, 0.9*(Uc) <= a1*x^2/2 - a3*x^4/4
<= 1.1*(Uc)]
x0 = 0.1; (* starting point for x[t] *) y0 = 0.1; (* starting point for
y[t] *)
proc = ItoProcess[ {\[DifferentialD]x[t] == y[t] \[DifferentialD]t,
\[DifferentialD]y[t] == (-G*y[t] - (a1*x[t] - a3*x[t]^3) - eps*b3b*y[t]^3)
\[DifferentialD]t + Sqrt[2*eps*G] \[DifferentialD]w[t]}, {t, x[t],
y[t], Boole[const[x[t], y[t]]]}, {{x, y}, {x0, y0}}, {t, 0}, w
\[Distributed] WienerProcess[]]
Exit time is found:
SeedRandom[3]
sim = RandomFunction[proc, {0, 1, 0.001}];
First@Select[sim[[2, 1, 1]], #[[4]] == 0 &]
that outputs {t, x[t], y[t], Boole[const[x[t], y[t]]]}



Issue:
The code above for the exit time only seems to produce non-zero exit times
for appropriately chosen initial conditions. If someone can elucidate as
to how to pick the appropriate $(x_0,y_0)$ such that there will be
sufficient/any non-zero exit time being produced then that would be
appreciative. I really want to keep my constraints as specified above in
const[x_, y_], but if there seems to be no hope in finding tractable
results then I wouldn't mind relaxing it.
Or, if there is another way to find the exit time for my particular system
then that would be appreciated as well (MATLAB is ok as well). I need to
find about 100 exit times for my further calculations.