Summary
Under appropriate initial data, solutions of the Kolmogorov equation \(u_{t = } \tfrac{1}{2}u_{xx} + f(u)\) 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].
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The research for this paper was done at the University of Wisconsin at Madison and supported in part by the National Science Foundation under Grant No. DMS-83-01080
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Bramson, M. Location of the travelling wave for the Kolmogorov equation. Probab. Th. Rel. Fields 73, 481–515 (1986). https://doi.org/10.1007/BF00324848
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DOI: https://doi.org/10.1007/BF00324848
Keywords
- Initial Data
- Stochastic Process
- Probability Theory
- Early Time
- General Formula