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Brownian motion with negative drift and convex level sets in space-time
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  • Published: September 1991

Brownian motion with negative drift and convex level sets in space-time

  • Christer Borell1 

Probability Theory and Related Fields volume 87, pages 403–409 (1991)Cite this article

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Summary

Suppose a, b, and μ are reals witha<b and consider the following diffusion equation

$$\begin{gathered} \frac{1}{2} \frac{{\partial ^2 u}}{{\partial x^2 }} + \mu \frac{{\partial u}}{{\partial x}} = \frac{{\partial u}}{{\partial t}} \hfill \\ u(x,0) = 0 \hfill \\ u(a,t) = 0 \hfill \\ u(b,t) = 1(a< x< b,t > 0). \hfill \\ \end{gathered} $$

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.

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References

  1. Borell, C.: Brownian motion in a convex ring and quasi-concavity. Commun. Math. Phys.86, 143–147 (1982)

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  2. Borell, C.: The Brunn-Minkowski inequality in Gauss space. Invent. math.30, 207–216 (1975)

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  3. 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

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  4. Harrison, J.M.: Brownian motion and stochastic flow systems. New York: Wiley 1985

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Author information

Authors and Affiliations

  1. Department of Mathematics, Chalmers University of Technology, S-41296, Göteborg, Sweden

    Christer Borell

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  1. Christer Borell
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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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  • Received: 10 August 1988

  • Revised: 31 July 1990

  • Issue Date: September 1991

  • DOI: https://doi.org/10.1007/BF01312218

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Keywords

  • Stochastic Process
  • Brownian Motion
  • Probability Theory
  • Diffusion Equation
  • Mathematical Biology
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