A Stochastic Stefan Problem


We consider a stochastic perturbation of the Stefan problem. The noise is Brownian in time and smoothly correlated in space. We prove existence and uniqueness and characterize the domain of existence.

This is a preview of subscription content, access via your institution.

Fig. 1


  1. 1.

    Barbu, V., Da Prato, G.: The two-phase stochastic Stefan problem. Probab. Theory Relat. Fields 124(4), 544–560 (2002)

    MATH  Article  Google Scholar 

  2. 2.

    Barbu, V., Da Prato, G., Röckner, M.: Existence of strong solutions for stochastic porous media equation under general monotonicity conditions. Ann. Probab. 37(2), 428–452 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Caffarelli, L., Salsa, S.: A Geometric Approach to Free Boundary Problems. Graduate Studies in Mathematics, vol. 68. American Mathematical Society, Providence (2005)

    Google Scholar 

  4. 4.

    Caffarelli, L.A., Lee, K.-A., Mellet, A.: Homogenization and flame propagation in periodic excitable media: the asymptotic speed of propagation. Commun. Pure Appl. Math. 59(4), 501–525 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Chow, P.L.: Stochastic Partial Differential Equations. Chapman & Hall/CRC, Boca Raton (2007)

    Google Scholar 

  6. 6.

    Da Prato, G., Röckner, M.: Invariant measures for a stochastic porous medium equation. In: Stochastic Analysis and Related Topics in Kyoto. Adv. Stud. Pure Math., vol. 41, pp. 13–29. Math. Soc. Japan, Tokyo (2004)

    Google Scholar 

  7. 7.

    Da Prato, G., Röckner, M.: Weak solutions to stochastic porous media equations. J. Evol. Equ. 4(2), 249–271 (2004)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Da Prato, G., Röckner, M., Rozovskii, B.L., Wang, F.-Y.: Strong solutions of stochastic generalized porous media equations: Existence, uniqueness, and ergodicity. Commun. Partial Differ. Equ. 31(1–3), 277–291 (2006)

    MATH  Article  Google Scholar 

  9. 9.

    Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)

    Google Scholar 

  10. 10.

    Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)

    Google Scholar 

  11. 11.

    Gaines, J.G.: Numerical experiments with S(P)DE’s. In: Stochastic Partial Differential Equations. London Math. Soc. Lecture Note Ser., vol. 216, pp. 55–71. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  12. 12.

    Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525–546 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991)

    Google Scholar 

  14. 14.

    Kim, J.U.: On the stochastic porous medium equation. J. Differ. Equ. 220(1), 163–194 (2006)

    MATH  Article  Google Scholar 

  15. 15.

    Kim, K., Mueller, C., Sowers, R.B.: A stochastic moving boundary value problem. Illinois J. Math. (2011), to appear

  16. 16.

    Lunardi, A.: An introduction to parabolic moving boundary problems. In: Functional Analytic Methods for Evolution Equations. Lecture Notes in Math., vol. 1855, pp. 371–399. Springer, Berlin (2004)

    Google Scholar 

  17. 17.

    McOwen, R.C.: Partial Differential Equations: Methods and Applications. Prentice-Hall, Upper Saddle (1996)

    Google Scholar 

  18. 18.

    Walsh, J.B.: An introduction to stochastic partial differential equations. In: École d’été de probabilités de Saint-Flour, XIV–1984. Lecture Notes in Math., vol. 1180, pp. 265–439. Springer, Berlin (1986)

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Kunwoo Kim.

Additional information

This work was supported by NSF grant DMS0705260. R.S. would like to thanks the Departments of Mathematics and Statistics of Stanford University for their hospitality in the Spring of 2010 during a sabbatical stay. The authors would like to thank the anonymous referee for his meticulous reading of the original version of this manuscript.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kim, K., Zheng, Z. & Sowers, R.B. A Stochastic Stefan Problem. J Theor Probab 25, 1040–1080 (2012). https://doi.org/10.1007/s10959-011-0392-1

Download citation


  • Stochastic partial differential equations
  • Stefan boundary condition
  • Spatially correlated noise

Mathematics Subject Classification (2000)

  • 60H15
  • 35R35
  • 80A22