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Formula for the solution of the problem of Brownian motion. (Russian. English summary) Zbl 0972.35113
The distribution of Brownian particles in position and time coordinates satisfies the Fokker-Planck equation, as it is well known. This equation has been investigated in many papers. In this paper, the author finds an explicit form of the solution of the equation \[ \sum^n_{i=1} {\partial^2w\over\partial u^2_i}- \sum^n_{i=1} u_i{\partial w\over\partial x_i}- {\partial w\over\partial t}= 0,\quad x\in \mathbb{R}^n_-,\;u\in\mathbb{R}^n,\;t>0, \] with \(w\) satisfying some conditions and with some boundary conditions in the half-space.
MSC:
35Q40 PDEs in connection with quantum mechanics
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
60J65 Brownian motion
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