Problem of the first passage time for <Emphasis Type="Italic">p</Emphasis>-adic diffusion

This work is a continuation of paper 1], where was considered analog of the problem of the first return for ultrametric diffusion. The main result of this paper consists in construction and investigation of stochastic quantity $ \tau _{B_r (a)} $ \tau _{B_r (a)} (ω), which has meaning of the first passage time into domain B _r (a) by trajectories of the Markov stochastic process ζ(t, ω).Markov stochastic process is given by distribution density f(x, t), x ∈ ℚ _p , t ∈ R ₊, which is solution of the Cauchy problem

$ \frac{\partial } {{\partial t}}f(x,t) = - D_x^\alpha f(x,t),f(x,0) = \Omega (\left| x \right|_p ). $ \frac{\partial } {{\partial t}}f(x,t) = - D_x^\alpha f(x,t),f(x,0) = \Omega (\left| x \right|_p ).
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