= Diffusion with hyperbolic tangent drift
{title2=$dX_t=\tanh X_t\,dt+dW_t$}
The scalar <stochastic differential equation> $dX_t=\tanh X_t\,dt+dW_t$ has a pathwise unique global strong solution, since its coefficients are globally Lipschitz. The positive martingale $e^{t/2}/\cosh X_t$ gives a <Girsanov theorem> change of measure under which $X_t-X_0$ is <Brownian motion>. For $X_0=x$, its transition density is
$$
p_t(x,z)=e^{-t/2}\frac{\cosh z}{\cosh x}(2\pi t)^{-1/2}e^{-(z-x)^2/(2t)}.
$$
This is a <Doob h-transform> with $h=\cosh$, and a mixture of $N(x+t,t)$ and $N(x-t,t)$ with weights $e^x/(2\cosh x)$ and $e^{-x}/(2\cosh x)$.
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