From Osher Doctorow
Meisner, Thorne, and Wheeler (MSW) in their 1973 "Gravitation" classic
had several routes to various major results, and here comes another
major route to Quantum Gravity and Electromagnetics via the Lotka-
Volterra and Super-Brownian Equations and Probable Influence/Causation
(PI).
Note first that PI is most closely related to the Riccati Differential
Equation from this thread and various of my previous threads:
1) dy/dt = A(t) + B(t)y + C(t)y^2
However, the Lotka-Volterra equations are quite close to the Riccati
Differential Equation and can be regarded as generalizations of the
latter, or special cases of generalizations of the latter. Their
continuous version, as for example in Wolfram "Lotka-Volterra
Equations", is:
2) dx/dt = Ax - Bxy
dy/dt = -Cy + Dxy, where A, B, C, D are positive constants
It turns out that the Lotka-Volterra equations have a stochastic
(probabilistic) version or generalization which turns out to be
asymptotically (in a limit) Super-Brownian Motion (see the latter as
keywords in arXiv and Front for the Mathematics ArXiv, where it has 26
papers). But Super-Brownian Motion relates to the Schrodinger
Equation with a one-point potential, the latter used in Quantum Theory
to describe singular electromagnetic effects on Quantum potentials,
e.g., a monograph by Albeveris et al in 1988 and afterward.
See Klaus Kleischmann, Carl Mueller, and Pascal Vogt, "On the large-
scale behavior of super-Brownian motion in three dimensions with a
single point source," respectively Weierstrass Institute for Applied
Analysis and Stochastics Berlin Germany, U. Rochester N.Y. USA, and
IFBAG Germany, math.PR/0607667 v2 26 Jul 2006, 6 pages. Also J.
Theodore Cox and Edwin A. Perkins, "Rescaled Lotka-Volterra models
converge to Super-Brownian-Motion," math.PR/0506591, which was
published in the prestigious Annals of Probability 2005 33(3),
904-957, 45 pages.
Osher Doctorow
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