Article,
Location of the travelling wave for the Kolmogorov equation
Probability Theory and Related Fields, 73 (4): 481--515 (November 1986)
Abstract
Under appropriate initial data, solutions of the Kolmogorov equation $$u_t = 12u_xx + f(u)$$ converge to travelling waves w λ (x), λ≧21/2, as t→∞. In the case λ>21/2, a general formula for the asymptotic position of the wave is known, as are formulas for certain cases of λ=21/2. The general formula for the case λ=21/2 given in 4 involves the behavior of the solution at earlier times and is typically not explicit enough for computation. Here, we give an improvement of this formula not requiring such information. The methodology involves use of sample path estimates for Brownian bridge and manipulation of certain formulas from 4.
