## Electronic Communications in Probability

### Improved Hölder continuity near the boundary of one-dimensional super-Brownian motion

Jieliang Hong

#### Abstract

We show that the local time of one-dimensional super-Brownian motion is locally $\gamma$-Hölder continuous near the boundary if $0<\gamma <3$ and fails to be locally $\gamma$-Hölder continuous if $\gamma >3$.

#### Article information

Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 28, 12 pp.

Dates
Accepted: 26 April 2019
First available in Project Euclid: 5 June 2019

https://projecteuclid.org/euclid.ecp/1559700464

Digital Object Identifier
doi:10.1214/19-ECP237

Mathematical Reviews number (MathSciNet)
MR3962478

Zentralblatt MATH identifier
07068652

#### Citation

Hong, Jieliang. Improved Hölder continuity near the boundary of one-dimensional super-Brownian motion. Electron. Commun. Probab. 24 (2019), paper no. 28, 12 pp. doi:10.1214/19-ECP237. https://projecteuclid.org/euclid.ecp/1559700464

#### References

• [1] R. Adler and M. Lewin. Local time and Tanaka formulae for super-Brownian motion and super stable processes. Stochastic Process. Appl., 41: 45–67, (1992).
• [2] M. Barlow, S. Evans and E. Perkins. Collision local times and measure-valued diffusions. Can. J. Math., 43: 897-938, (1991).
• [3] M. E. Caballero, A. Lambert, and G. U. Bravo. Proof(s) of the Lamperti representation of continuous-state branching processes. Probab. Surv., 6: 62–89, (2009).
• [4] J. Hong. Renormalization of local times of super-Brownian motion. Electron. J. Probab., 23: no. 109, 1–45, (2018).
• [5] N. Konno and T. Shiga. Stochastic partial differential equations for some measure-valued diffusions. Probab. Theory Relat. Fields, 79: 201–225, (1988).
• [6] S. Krone. Local times for superdiffusions. Ann. Probab., 21 (b): 1599–1623, (1993).
• [7] J.F. Le Gall. Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics, ETH, Zurich. Birkhäuser, Basel (1999).
• [8] J.F. Le Gall. The Hausdorff Measure of the Range of Super-Brownian Motion. In: Bramson M., Durrett R. (eds) Perplexing Problems in Probability. Progress in Probability, vol 44. Birkhäuser, Boston (1999).
• [9] M. Merle. Local behavior of local times of super-Brownian motion. Ann. I. H. Poincaré–PR, 42: 491–520, (2006).
• [10] P. Morters and Y. Peres. Brownian Motion. Cambridge University Press, Cambridge (2010).
• [11] L. Mytnik and E. Perkins. Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case. Probab. Theory Relat. Fields, 149: 1–96, (2011).
• [12] L. Mytnik and E. Perkins. The dimension of the boundary of super-Brownian motion. arXiv:1711.03486. To appear in Prob. Th. Rel Fields.
• [13] L. Mytnik, E. Perkins and A. Sturm. On pathwise uniqueness for stochastic heat equations with non-Lipschitz coefficients. Ann. Probab., 34: 1910–1959, (2006).
• [14] E.A. Perkins. Dawson-Watanabe Superprocesses and Measure-valued Diffusions. Lectures on Probability Theory and Statistics, no. 1781, Ecole d’Eté de Probabilités de Saint Flour 1999. Springer, Berlin (2002).
• [15] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer, Berlin (1994).
• [16] S. Sugitani. Some properties for the measure-valued branching diffusion processes. J. Math. Soc. Japan, 41: 437–462, (1989).