The Annals of Applied Probability

Smooth density field of catalytic super-Brownian motion

Klaus Fleischmann and Achim Klenke

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Given an (ordinary) super-Brownian motion (SBM) $\varrho$ on $\mathbf{R}^d$ of dimension $d = 2, 3$, we consider a (catalytic) SBM $X^{\varrho}$ on $\mathbf{R}^d$ with "local branching rates" $\varrho_s(dx)$. We show that $X_t^{\varrho}$ is absolutely continuous with a density function $\xi_t^{\varrho}$, say. Moreover, there exists a version of the map $(t, z) \mapsto \xi_t^{\varrho}(z)$ which is $\mathscr{C}^{\infty}$ and solves the heat equation off the catalyst $\varrho$; more precisely, off the (zero set of) closed support of the time-space measure $ds\varrho_s(dx)$. Using self-similarity, we apply this result to give the following answer to an open problem on the long-term behavior of $X^{\varrho}$ in dimension $d = 2$: If $\varrho$ and $X^{\varrho}$ start with a Lebesgue measure, then does $X_T^{\varrho}$ converge (persistently) as $T \to \infty$ toward a random multiple of Lebesgue measure?

Article information

Ann. Appl. Probab., Volume 9, Number 2 (1999), 298-318.

First available in Project Euclid: 21 August 2002

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Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G57: Random measures 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Superprocess persistence absolutely continuous states time-space gaps of super-Brownian motion smooth density field diffusive measures


Fleischmann, Klaus; Klenke, Achim. Smooth density field of catalytic super-Brownian motion. Ann. Appl. Probab. 9 (1999), no. 2, 298--318. doi:10.1214/aoap/1029962743.

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  • [1] Barlow, M. T., Evans, S. N. and Perkins, E. A. (1991). Collision local times and measurevalued processes. Canad. J. Math. 43 897-938.
  • [2] Dawson, D. A. (1993). Measure-valued Markov processes. ´Ecole d' ´Et´e de Probabilit´es de Saint Flour XXI Lecture Notes in Math. 1541 1-260. Springer, Berlin.
  • [3] Dawson, D. A. and Fleischmann, K. (1995). Super-Brownian motions in higher dimensions with absolutely continuous measure states. J. Theoret. Probab. 8 179-206.
  • [4] Dawson, D. A. and Fleischmann, K. (1997). A continuous super-Brownian motion in a super-Brownian medium. J. Theoret. Probab. 10 213-276.
  • [5] Dawson, D. A. and Fleischmann, K. (1997). Longtime behavior of a branching process controlled by branching cataly sts. Stochastic Process. Appl. 71 241-257.
  • [6] Dawson, D. A., Fleischmann, K. and Roelly, S. (1991). Absolute continuity for the measure states in a branching model with cataly sts. Progr. Probab. 24 117-160.
  • [7] Dawson, D. A. and Hochberg, K. J. (1979). The carrying dimension of a stochastic measure diffusion. Ann. Probab. 7 693-703.
  • [8] Dawson, D. A., Iscoe, I. and Perkins, E. A. (1989). Super-Brownian motion: path properties and hitting probabilities. Probab. Theory Related Fields 83 135-205.
  • [9] Delmas, J.-F. (1996). Super-mouvement brownien avec cataly se. Stochastics Stochastics Rep. 58 303-347.
  • [10] Dy nkin, E. B. (1991). Branching particle sy stems and superprocesses. Ann. Probab. 19 1157-1194.
  • [11] Dy nkin, E. B. (1993). Superprocesses and partial differential equations. Ann. Probab. 21 1185-1262.
  • [12] Etheridge, A. M. and Fleischmann, K. (1998). Persistence of a two-dimensional superBrownian motion in a cataly tic medium. Probab. Theory Related Fields 110 1-12.
  • [13] Evans, S. N. and Perkins, E. A. (1991). Absolute continuity results for superprocesses with some applications. Trans. Amer. Math. Soc. 325 661-681.
  • [14] Evans, S. N. and Perkins, E. A. (1994). Measure-valued branching diffusions with singular interactions. Canad. J. Math. 46 120-168.
  • [15] Fleischmann, K. and G¨artner, J. (1986). Occupation time processes at a critical point. Math. Nachr. 125 275-290.
  • [16] Fleischmann, K. and LeGall, J.-F. (1995). A new approach to the single point cataly tic super-Brownian motion. Probab. Theory Related Fields 102 63-82.
  • [17] Greven, A., Klenke, A. and Wakolbinger, A. (1997). The longtime behaviour of branching random walk in a cataly tic medium. Preprint, Univ. Erlangen-N ¨urnberg.
  • [18] Konno, N. and Shiga, T. (1988). Stochastic partial differential equations for some measurevalued diffusions. Probab. Theory Related Fields 79 201-225.
  • [19] Le Gall, J.-F. (1994). A lemma on super-Brownian motion with some applications. Progr. Probab. 34 237-251.
  • [20] Perkins, E. (1989). The Hausdorff measure of the closed support of super-Brownian motion. Ann. Inst. H. Poincar´e Probab. Statist. 25 205-224.
  • [21] Perkins, E. A. (1990). Polar sets and multiple points for super-Brownian motion. Ann. Probab. 18 453-491.