The Annals of Probability

Nonuniqueness for a parabolic SPDE with $\frac{3}{4}-\varepsilon $-Hölder diffusion coefficients

Carl Mueller, Leonid Mytnik, and Edwin Perkins

Full-text: Open access

Abstract

Motivated by Girsanov’s nonuniqueness examples for SDEs, we prove nonuniqueness for the parabolic stochastic partial differential equation (SPDE) \[\frac{\partial u}{\partial t}=\frac{\Delta}{2}u(t,x)+\bigl|u(t,x)\bigr|^{\gamma}\dot{W}(t,x),\qquad u(0,x)=0.\] Here $\dot{W}$ is a space–time white noise on $\mathbb{R}_{+}\times\mathbb{R}$. More precisely, we show the above stochastic PDE has a nonzero solution for $0<\gamma<3/4$. Since $u(t,x)=0$ solves the equation, it follows that solutions are neither unique in law nor pathwise unique. An analogue of Yamada–Watanabe’s famous theorem for SDEs was recently shown in Mytnik and Perkins [Probab. Theory Related Fields 149 (2011) 1–96] for SPDE’s by establishing pathwise uniqueness of solutions to \[\frac{\partial u}{\partial t}=\frac{\Delta}{2}u(t,x)+\sigma \bigl(u(t,x)\bigr)\dot{W}(t,x)\] if $\sigma$ is Hölder continuous of index $\gamma>3/4$. Hence our examples show this result is essentially sharp. The situation for the above class of parabolic SPDE’s is therefore similar to their finite dimensional counterparts, but with the index $3/4$ in place of $1/2$. The case $\gamma=1/2$ of the first equation above is particularly interesting as it arises as the scaling limit of the signed mass for a system of annihilating critical branching random walks.

Article information

Source
Ann. Probab., Volume 42, Number 5 (2014), 2032-2112.

Dates
First available in Project Euclid: 29 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1409319473

Digital Object Identifier
doi:10.1214/13-AOP870

Mathematical Reviews number (MathSciNet)
MR3262498

Zentralblatt MATH identifier
1301.60080

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 35K05: Heat equation

Keywords
Heat equation white noise stochastic partial differential equations

Citation

Mueller, Carl; Mytnik, Leonid; Perkins, Edwin. Nonuniqueness for a parabolic SPDE with $\frac{3}{4}-\varepsilon $-Hölder diffusion coefficients. Ann. Probab. 42 (2014), no. 5, 2032--2112. doi:10.1214/13-AOP870. https://projecteuclid.org/euclid.aop/1409319473


Export citation

References

  • Burdzy, K., Mueller, C. and Perkins, E. A. (2010). Nonuniqueness for nonnegative solutions of parabolic stochastic partial differential equations. Illinois J. Math. 54 1481–1507.
  • Burkholder, D. L. (1973). Distribution function inequalities for martingales. Ann. Probab. 1 19–42.
  • Knight, F. B. (1981). Essentials of Brownian Motion and Diffusion. Mathematical Surveys 18. Amer. Math. Soc., Providence, RI.
  • Konno, N. and Shiga, T. (1988). Stochastic partial differential equations for some measure-valued diffusions. Probab. Theory Related Fields 79 201–225.
  • Kurtz, T. G. (2007). The Yamada–Watanabe–Engelbert theorem for general stochastic equations and inequalities. Electron. J. Probab. 12 951–965.
  • Meyer, P.-A. (1966). Probability and Potentials. Blaisdell Publishing, Waltham, MA.
  • Mueller, C. and Perkins, E. A. (1992). The compact support property for solutions to the heat equation with noise. Probab. Theory Related Fields 93 325–358.
  • Mytnik, L. (1998). Weak uniqueness for the heat equation with noise. Ann. Probab. 26 968–984.
  • Mytnik, L. and Perkins, E. (2011). Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: The white noise case. Probab. Theory Related Fields 149 1–96.
  • Mytnik, L., Perkins, E. and Sturm, A. (2006). On pathwise uniqueness for stochastic heat equations with non-Lipschitz coefficients. Ann. Probab. 34 1910–1959.
  • Perkins, E. (2002). Dawson–Watanabe superprocesses and measure-valued diffusions. In Lectures on Probability Theory and Statistics (Saint-Flour, 1999). Lecture Notes in Math. 1781 125–324. Springer, Berlin.
  • Reimers, M. (1989). One-dimensional stochastic partial differential equations and the branching measure diffusion. Probab. Theory Related Fields 81 319–340.
  • Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin.
  • Rogers, L. C. G. and Williams, D. (1987). Diffusions, Markov Processes, and Martingales. Vol. 2: Itô Calculus. Wiley, New York.
  • Shiga, T. (1988). Stepping stone models in population genetics and population dynamics. In Stochastic Processes in Physics and Engineering (Bielefeld, 1986). Math. Appl. 42 345–355. Reidel, Dordrecht.
  • Shiga, T. (1994). Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Canad. J. Math. 46 415–437.
  • Viot, M. (1975). Méthodes de compacité et de monotonie compacitè pour les équations aux dérivees partielles stochastiques. Thèse, Univ. de Paris.
  • Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École d’été de probabilités de Saint-Flour, XIV-1984 (P. L. Hennequin, ed.). Lecture Notes in Math. 1180 265–439. Springer, Berlin.
  • Yamada, T. and Watanabe, S. (1971). On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 155–167.