Geodesic
On a~probability problem for a~one-dimensional heat equation
Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 4, pp. 727-729

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The system (2) for random amplitudes $W_i(t)$ (where $f(t)$ is the derivative of a Wiener process) was considered in [1] in connection with the stochastic heat equation (1) and the two following assertions were obtained: (a) a solution of the system (2) is a random (normal) element in the Hilbert space $l_2$ for every $t>0$; (b) almost all solutions $\{W_i(t)\}$ are rapidly decreasing sequences for every $t>0$. In the present note a simple proof of the assertion (a) is given and the assertion (b) is shown to be wrong. 
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     author = {N. N. Vakhania},
     title = {On a~probability problem for a~one-dimensional heat equation},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {727--729},
     publisher = {mathdoc},
     volume = {12},
     number = {4},
     year = {1967},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1967_12_4_a10/}
}
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N. N. Vakhania. On a~probability problem for a~one-dimensional heat equation. Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 4, pp. 727-729. http://geodesic.mathdoc.fr/item/TVP_1967_12_4_a10/