Geodesic
On the evolution of distributed systems when there is a~fluctuation of the density on the boundary
Teoriâ veroâtnostej i ee primeneniâ, Tome 10 (1965) no. 4, pp. 736-741

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A dynamical system is considered which is described by a parabolic equation in a circle of length $2\pi$ when acted upon by an undistributed stochastic source with a power $\dot\pi(t)$ (the derivative of Poisson's process): $$ \frac{\partial W(x,t)}{\partial t}-D^2\frac{\partial^2W(x,t)}{\partial x^2}=\delta(x)\dot\pi(t). $$ The characteristic functional for this system which defines a countable additive measure iii the phase space is constructed. It is proved that almost all $W(x)$ are infinitely differentiable. This measure is not quasi-invariant. 
@article{TVP_1965_10_4_a10,
     author = {A. A. Beilinson},
     title = {On the evolution of distributed systems when there is a~fluctuation of the density on the boundary},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {736--741},
     publisher = {mathdoc},
     volume = {10},
     number = {4},
     year = {1965},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1965_10_4_a10/}
}
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A. A. Beilinson. On the evolution of distributed systems when there is a~fluctuation of the density on the boundary. Teoriâ veroâtnostej i ee primeneniâ, Tome 10 (1965) no. 4, pp. 736-741. http://geodesic.mathdoc.fr/item/TVP_1965_10_4_a10/