Abstract
The model studied concerns a simple first-order hyperbolic system. The solutions in which one is most interested have discontinuities which persist for all time, and therefore need to be interpreted as weak solutions. We demonstrate existence and uniqueness for such weak solutions, identifying a canonical `exact' solution which is everywhere defined. The direct method used is guided by the theory of measure-valued diffusions. The method is more effective than the method of characteristics, and has the advantage that it leads immediately to the McKean representation without recourse to Itô's formula. We then conduct computer studies of our model, both by integration schemes (which do use characteristics) and by `random simulation'.
Highlights
After a change of the original notation with u replacing 1 − u, the FKPP equation (Fisher [6], Kolmogorov, Petrovskii & Piskunov [9]) reads: ∂u ∂t = ∂2u 2 ∂x2 + u2− u, u(0, x) = f (x).McKean [12, 13] showed that one can prove existence and uniqueness results for certain [0, 1]valued solutions by using martingales to describe u as an explicit functional of a certain branching Brownian motion
The direct method used is guided by the theory of measure-valued diffusions, MVDs,. (We stress that no knowledge of MVD theory is assumed here.) The method is more effective than the method of characteristics, and has the advantage that it leads immediately to the McKean representation without recourse to Ito’s formula
We know that v(t, x) is the probability that all particles are to the right of 0 at time t
Summary
After a change of the original notation with u replacing 1 − u, the FKPP equation (Fisher [6], Kolmogorov, Petrovskii & Piskunov [9]) reads:. McKean [12, 13] showed that one can prove existence and uniqueness results for certain [0, 1]valued solutions by using martingales to describe u as an explicit functional of a certain branching Brownian motion. Two recent papers — Champneys, Harris, Toland, Warren & Williams [2] and Lyne [10] — have each presented both analytic and probabilistic studies of simple extensions of the FKPP equation. These papers used the probabilist’s golden rule that Ito’s formula leads to martingales (see, for example, Rogers & Williams [17]).
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