Distributional Solution of the Ornstein-Uhlenbeck SDE

For now we will examine the special case of a scalar equation 

We saw before that we could think of the simple SDE initial value problem (6.8) as having a solution defined by a certain random process. In the same way we would like to obtain, for the Ornstein-Uhlenbeck SDE (6.19), an explicit stochastic process which is in some sense equivalent to solving the SDE. Multiply both sides of (6.19) by the integrating factor exp(yf), and observe that 

Using Proposition 6.3, we know that for a Wiener process W(f), Y(f) exp(y5) dW(x) is a Gaussian random variable with mean zero and variance 

Let CT = V2ykBTm, then for large t the exponentials of the form e tend to zero, and we have 

If we think of X(t) as not the position, but actually the momentum, then the Ornstein-Uhlenbeck equation samples, in the long term, the Gibbs-Boltzmann distribution associated to a single particle in a thermal bath at temperature T. In recognition of this, from this point forward, we will replace X with p as the variable in the Ornstein-Uhlenbeck equation. For a system with configurational variables 

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