= Stationary Gaussian forward curve
{title2=$K(x,y)=\int_0^\infty\langle\sigma(x+s),\sigma(y+s)\rangle ds$}
If the displayed covariance is finite and the mean obeys $m'(x)+A(x)=0$, initialize the <Musiela forward-curve equation> with the corresponding <Gaussian random field>, independently of future noise. The resulting curve is stationary in time. Indeed shifting the initial covariance contributes the integral over $[t,\infty)$ and fresh noise contributes the integral over $[0,t]$, so their sum is $K$. A deterministic initial curve generally is not stationary, and nondecaying volatility may make $K(x,x)$ infinite. Exponentially decaying finite-factor volatilities give stationary <Ornstein-Uhlenbeck processes>.
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