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2) defining the characteristic function ¢(z) will converge in the domain common to the convergence strips of ¢1 (z) and ¢ 11 (z). ¢(z) will then be analytic in the interior of this convergence strip. 8) do not converge beyond V = 0. 8) converge beyond V = 0 and the convergence strip contains the real z axis in its interior. The boundaries of the convergence strip for ¢(z) are not in general the natural boundaries of the characteristic function which will have an analytic continuation beyond the boundaries of the strip.

The Wiener-Levy process is the only homogeneous process with this property, terms XO + vt apart. 19) is of interest in its own right since it describes a simple type of Brownian motion with positive increments. 4) with XJ(O) 0, and XD (t) is a Wiener-Levy process. The corresponding transition distribution of X(t) will be the convolution of Fo(x) and the transition distributions for the other components. When the Wiener-Levy component is present, the distribution of X(t) will inherit the absolute continuity for t > 0 of that for XD (t).

When v * 0, a similar simple interpretation is available. Instead of observing the rate of change at some fixed x, we may have an observer moving with velocity v monitoring the density fvCx 0 + vt, t) at the position Yt = x 0 + vt. The rate of change of density at the moving position of the observer is fttv(Yi,t), and by the above reasoning this must equal -v fv(y t ,t) + v f fv(x', t) x a(yt-x')dx'. 14). The character of the operator I!.. = ( l.. +i_) as a moving differentiation in time is familiar in Dt at ax hydrodynamics.

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