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Let $W_t$ denote the Wiener process and let

$$ dX_t = a(t, X_t) dt + b(t, X_t) dW_t $$

be an one dimensional Ito SDE (stochastic differential equation, see Ito stochastic calculus).

A hitting time $\tau_H$ for a suitable subset $H \subset \mathbb{R}$ is a random variable depending on the sample paths of $X_t$.

I'm interested in a concrete SDE that can be used to explain stochastic resonance (there is also a page about this on the Azimuth project of John Baez, but the Wiki is down at the moment, unfortunately).

This would be $$ dX_t = (X_t - X_t^3 + A \; sin(t + \phi)) \; dt + C \; dW_t $$ This equation has three real positive parameters $A, \phi, C \in \mathbb{R}_+$. The initial condition is $X_0 = 1$.

What is known about the hitting time $\tau_{[- \infty, 0]}$? (That is the time when the process first crosses 0, coming from 1).

I'm interested in all kinds of analytic results that one could e.g. use to crosscheck results from numerical simulations. I'm also interested in results known for simpler models (or more general ones, of course).

First example: Are there any results if we simplify the problem by eliminating the explicit time dependency, setting $A = 0$?

Second example: I think I once read about the asymtotic expansion of the hitting time of the Ornstein-Uhlenbeck process for small t (for getting absorbed at a prescribed point) , but could not find any reference.

The reason for this question was that John Baez asked about proofs for the existence of stochastic resonance, like a proof for the existence of an optimal value of the drift constant $C$, so if my question is misguided in the light of this goal, please tell me :-)

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1 Answer

I just discovered some interesting theorems on stochastic resonance here:

Peter Imkeller, Energy balance models - viewed from stochastic dynamics.

In particular, Theorem 5.4 near the end seems to be a step towards proving the existence of stochastic resonance in the model you describe.

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Wow, that was quick! As Peter Imkeller himself points out, that paper leaves some questions unanswered, so I won't mark the question as being answered, but it is definitely a step forward. (And now I know a theorem that justifies the Kramers rate! Cool.) We should take a look at Freidlin, M.I.; Wentzell, A.D., "Random perturbations of dynamical systems." –  Tim van Beek Nov 21 '10 at 20:40
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