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Hmmm... .. Thats a wicked question!!
Dunno.. But lets think together, without maths
Brownian motion is due to:
1. Thermal energy of the particles
2. Continuous collisions between particles and particle-cavity boundaries
Lets take the velocity of the particles:
If the temperature is constant, then the velcoity of the particles ought to have a Gaussian PDF centered at the velocity corresponding to the kinteic energy of the molecules which are thermally active.
If no external perturbation is applied on the system once at steady state, there is no reason why this velcity pdf could change. So most probably this system is stationary in the wide sense. Just like thermal noise in electrical components (I guess thats the case). IF a Gaussian system is stationary in the wide sense, then its stationary in the strict sense.
What do you think?? I'm not quite sure of the above text, but thats what I thought of. It might be correct and it might be wrong. Please if you find any flaw or any error. I'ld be happy to talk about this more thoroughly.
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.. Thats a wicked question!! 