Occupation time large deviations for critical branching Brownian motion, super-Brownian motion and related processes
It is known [\textit{I. Iscoe}, Probab. Theory Relat. Fields 71, No. 1, 85-116 (1986; Zbl 0555.60034)] that for 3-dimensional super-Brownian motion \(\mu_t\) the occupation time fluctuations \[ T^{-3/4} \int^T_0(\langle\mu_s, h\rangle- \langle\lambda, h\rangle) ds \] converge in distribution as \(T\to\infty\) towards perfectly correlated centered Gaussian random variables with variance proportional to the Lebesgue integral \(\langle\lambda, h\rangle\) of the test function \(h\). In particular, for functions \(h\) with \(\langle\lambda, h\rangle= 0\), the normed occupation time \(T^{-3/4} \int^T_0\langle\mu_s, h\rangle ds\) converges to zero as \(T\to\infty\). The large deviations of this normed occupation time are investigated. The authors prove the large deviation principle \[ \lim_{T\to\infty} T^{-1/2}\log P\Biggl(T^{- 3/4} \int^T_0 \langle\mu_s, h\rangle ds\geq x\Biggr)= \Lambda^*_h(x), \] where \(\Lambda^*_h(x)\) is expressed in terms of the Legendre transform of the moment generating function of \(L^0_T\), the super-Browian local time up to time one. This settles a conjecture of \textit{T.-Y. Lee} and \textit{B. Rémillard} [Ann. Probab. 23, No. 4, 1755-1771 (1995; Zbl 0852.60004)]. The proof is based on an analysis of the rescaled moment generating functions of the occupation time, using Dynkin's graphical moment formula. As a local version of their result, the authors also prove a large deviation principle for the rescaled difference \(T^{-3/4}(L^x_t- L^0_T)\) of two local times. In addition, they derive from their large deviation principle a central limit theorem which states that \(T^{-1/2} \int^T_0 \langle\mu_s, h\rangle ds\) converges in distribution towards a centered Gaussian random variable with variance \[ -{1\over \pi} \iint|y- z|h(y) h(z) dy dz= \langle\lambda,(Gh)^2\rangle. \] The integral of the squared potential \((Gh)^2\) is finite only due to the assumption \(\langle\lambda, h\rangle= 0\); this property of \(h\) also guarantees that the ``global parts'' of the occupation time fluctuations, which (as stated above) grow like \(T^{3/4}\), cancel in the limit. Also, corresponding results for symmetric \(\beta\)-stable motion in dimension \(d\in(2(\beta- 1),2\beta)\), and for branching particle systems, are proved. For the latter, in addition to the term \(\langle\lambda, (Gh)^2\rangle\) the term \(2\langle\lambda, Gh\rangle\) also appears. This term comes from the contribution to the occupation time of those particles which are mutually related ``in direct line'' -- a contribution which vanishes in the superprocess limit.
scientific article; zbMATH DE number 1355248