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**A super Ornstein-Uhlenbeck process interacting with its center of mass.**
*(English)*
Zbl 1279.60110

This paper treats a supercritical interacting measure-valued diffusion process \(X_t\). This model can describe an interaction between representative particles that are attracted to, or repelled from, the center of mass. The author proposes a simplest model of this kind, named a super Ornstein-Uhlenbeck (SOU) process \(X'_t\) with attractor (or repeller) given by the center of mass, and shows several limit theorems for those superprocesses.

More precisely, it is proved that on the extinction set \(S^c\), the normalized process \(\tilde{X}_t\) in both the attractive and repelling cases converges to a random point almost surely as \(t\) approaches the extinction time \(\eta\). This result is an analogue of the result obtained by R. Tribe [Ann.Probab.20, No. 1, 286–311, (1992; Zbl 0749.60046)]. The main limit results of this paper consists of the following two theorems. The symbol \(d\) denotes the Vasserstein metric on the space \(M_F( {\mathbb R}^d)\) of finite measures on \({\mathbb R}^d\).

{Theorem 1.} On the survival set \(S\), in the attractive case, \[ d ( \tilde{X}_t, P_{\infty}) \to 0, \quad \text{a.s.} \tag{1} \] where \(\tilde{X}_t\) is the mass normalized interacting measure-valued process and \(P_t\) is the semigroup of an Ornstein-Uhlenbeck (OU) process with attraction to 0 at rate \(\gamma > 0\).

This means that the mass normalized process \(\tilde{X}_t\) converges to the stationary distribution of the OU process, centered at the limiting value of its center of mass.

{Theorem 2.} Under the same setting, \[ d ( \tilde{X}'_t, P_{\infty}^{ \bar{Y}_{\infty} } ) \to 0, \quad \text{a.s.} \tag{2} \] where \(\tilde{X}'_t\) is the normalized SOU process and \(P_{\infty}^{ \bar{Y}_{\infty} }\) is the OU semigroup at infinity with the origin shifted to the center of mass \(\bar{Y}_{\infty}\) at infinity.

This result indicates that \(\tilde{X}'_t\) converges almost surely to a Gaussian random variable, and is an extension of the result by J. Engländer and A. Winter [Ann.Inst.Henri Poincaré, Probab.Stat.42, No. 2, 171–185 (2006; Zbl 1093.60058)].

On the other hand, another peculiar feature of this paper consists in the usage of historical stochastic calculus of Perkins applied to SOU process with attraction, (cf. [E. Perkins, On the martingale problem for interactive measure-valued branching diffusions. Mem. Am. Math. Soc. 549 (1995; Zbl 0823.60071)]. Lastly, the author discusses the almost surely convergence of the center of mass in the repelling setting as well. However, because the repulsion condition posed is too strong, a conjecture about this convergence result is given.

For other related works, see, e.g. [J. Englander, Electron.J.Probab.15, 1938–1970 (2010; Zbl 1226.60118)] for a similar type superprocess, and [H. S. Gill, Stochastic Process.Appl.119, No. 2, 3981–4003 (2009; Zbl 1193.60083)] for an application of historical stochastic calculus of Perkins.

More precisely, it is proved that on the extinction set \(S^c\), the normalized process \(\tilde{X}_t\) in both the attractive and repelling cases converges to a random point almost surely as \(t\) approaches the extinction time \(\eta\). This result is an analogue of the result obtained by R. Tribe [Ann.Probab.20, No. 1, 286–311, (1992; Zbl 0749.60046)]. The main limit results of this paper consists of the following two theorems. The symbol \(d\) denotes the Vasserstein metric on the space \(M_F( {\mathbb R}^d)\) of finite measures on \({\mathbb R}^d\).

{Theorem 1.} On the survival set \(S\), in the attractive case, \[ d ( \tilde{X}_t, P_{\infty}) \to 0, \quad \text{a.s.} \tag{1} \] where \(\tilde{X}_t\) is the mass normalized interacting measure-valued process and \(P_t\) is the semigroup of an Ornstein-Uhlenbeck (OU) process with attraction to 0 at rate \(\gamma > 0\).

This means that the mass normalized process \(\tilde{X}_t\) converges to the stationary distribution of the OU process, centered at the limiting value of its center of mass.

{Theorem 2.} Under the same setting, \[ d ( \tilde{X}'_t, P_{\infty}^{ \bar{Y}_{\infty} } ) \to 0, \quad \text{a.s.} \tag{2} \] where \(\tilde{X}'_t\) is the normalized SOU process and \(P_{\infty}^{ \bar{Y}_{\infty} }\) is the OU semigroup at infinity with the origin shifted to the center of mass \(\bar{Y}_{\infty}\) at infinity.

This result indicates that \(\tilde{X}'_t\) converges almost surely to a Gaussian random variable, and is an extension of the result by J. Engländer and A. Winter [Ann.Inst.Henri Poincaré, Probab.Stat.42, No. 2, 171–185 (2006; Zbl 1093.60058)].

On the other hand, another peculiar feature of this paper consists in the usage of historical stochastic calculus of Perkins applied to SOU process with attraction, (cf. [E. Perkins, On the martingale problem for interactive measure-valued branching diffusions. Mem. Am. Math. Soc. 549 (1995; Zbl 0823.60071)]. Lastly, the author discusses the almost surely convergence of the center of mass in the repelling setting as well. However, because the repulsion condition posed is too strong, a conjecture about this convergence result is given.

For other related works, see, e.g. [J. Englander, Electron.J.Probab.15, 1938–1970 (2010; Zbl 1226.60118)] for a similar type superprocess, and [H. S. Gill, Stochastic Process.Appl.119, No. 2, 3981–4003 (2009; Zbl 1193.60083)] for an application of historical stochastic calculus of Perkins.

Reviewer: Isamu Dôku (Saitama)

### Keywords:

superprocess; interacting measure-valued diffusion process; Ornstein-Uhlenbeck process; center of mass; law of large numbers### References:

[1] | Engländer, J. (2010). The center of mass for spatial branching processes and an application for self-interaction. Electron. J. Probab. 15 1938-1970. · Zbl 1226.60118 |

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[11] | Polyanin, A. D. and Manzhirov, A. V. (2008). Handbook of Integral Equations , 2nd ed. Chapman & Hall/CRC, Boca Raton, FL. · Zbl 1154.45001 |

[12] | Roberts, G. O. and Tweedie, R. L. (2000). Rates of convergence of stochastically monotone and continuous time Markov models. J. Appl. Probab. 37 359-373. · Zbl 0979.60060 · doi:10.1239/jap/1014842542 |

[13] | Rogers, L. and Williams, D. (1985). Diffusions , Markov Processes and Martingales . Cambridge Univ. Press, Cambridge. · Zbl 0826.60002 |

[14] | Tribe, R. (1992). The behavior of superprocesses near extinction. Ann. Probab. 20 286-311. · Zbl 0749.60046 · doi:10.1214/aop/1176989927 |

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