Summary
Suppose a, b, and μ are reals witha<b and consider the following diffusion equation
If μ≦0, we prove that all the level sets {u≧p} (0<p<1) are convex. The special case μ=0 is well-known [1]. The present extension is mainly motivated by our interest in the Brownian exponential martingale. Actually, as in [1] the mathematical results of this paper are given for several space dimensions.
References
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Borell, C.: Analytic and empirical evidences of isoperimetric processes. (Progress in Probability vol. 20, pp. 13–40) In: Haagerup, U., Hoffmann-Jørgensen, J., Nielsen, N.J. (eds.) Probability in Banach spaces 6. Boston Basel Berlin: Birkhäuser 1990
Harrison, J.M.: Brownian motion and stochastic flow systems. New York: Wiley 1985
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Borell, C. Brownian motion with negative drift and convex level sets in space-time. Probab. Th. Rel. Fields 87, 403–409 (1991). https://doi.org/10.1007/BF01312218
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DOI: https://doi.org/10.1007/BF01312218
Keywords
- Stochastic Process
- Brownian Motion
- Probability Theory
- Diffusion Equation
- Mathematical Biology