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Supercritical super-Brownian motion with a general branching mechanism and travelling waves. (English. French summary) Zbl 1267.60094
This paper treats, in a probabilistic manner, the classical problem of existence, uniqueness and asymptotic behaviour of monotone solutions to the travelling wave equation associated with the parabolic semigroup equation of a super-Brownian motion $$X_t$$ with a general branching mechanism $$\psi$$. The leading philosophy is strongly due to the work done by the first author’s [Ann. Inst. Henri Poincaré, Probab. Stat. 40, No. 1, 53–72 (2004; Zbl 1042.60057)] for branching Brownian motions and the travelling wave solution to the KPP equation.
More precisely, let $$X = \{ X_t ; t \geq 0 \}$$ denote a one-dimensional $$\psi$$-super-Brownian motion with general branching mechanism $\psi(\lambda) = - \alpha \lambda + \beta \lambda^2 + \int_{(0, \infty)} ( e^{- \lambda x} - 1 + \lambda x) \nu(dx) \quad \text{for } \lambda \geq 0, \tag{1}$ where $$\alpha = - \psi'(0+) \in (0, \infty)$$, $$\beta \geq 0$$ and $$\nu$$ is a measure concentrated on $$(0, \infty)$$ satisfying $$\int_{(0, \infty)} ( x \wedge x^2) \nu(dx)<\infty$$, and let $${\mathcal M}_F({\mathbb R})$$ be the space of finite measures on $${\mathbb R}$$. Note that $$X$$ is an $${\mathcal M}_F({\mathbb R})$$-valued Markov process under $$\operatorname{P}_{\mu}$$, where $$\operatorname{P}_{\mu}$$ is the law of $$X$$ with initial configuration $$\mu \in {\mathcal M}_F({\mathbb R})$$. For $$f \in C_b^+({\mathbb R})$$ (the space of finite, uniformly bounded, continuous functions on $${\mathbb R}$$) and $$\mu \in {\mathcal M}_F({\mathbb R})$$, define $$\langle f, \mu \rangle=\int_{{\mathbb R}} f(x) \mu(dx)$$, and accordingly write $$\| \mu \|=\langle 1, \mu \rangle$$. The existence of the superprocess $$X_t$$ is guaranteed by results of E. B. Dynkin [Ann. Probab. 21, No. 3, 1185–1262 (1993; Zbl 0806.60066)]. The characterization of the superprocess $$X_t$$ is given by the following. For all $$f \in C_b^+({\mathbb R})$$ and $$\mu \in {\mathcal M}_F({\mathbb R})$$, $- \log \operatorname{E}_{\mu} ( e^{- \langle f, X_t \rangle}) = \int_{{\mathbb R}} u_f(x,t) \mu(dx), \quad t > 0, \tag{2}$ where $$u_f(x,t)$$ is the unique positive solution to the evolution equation $\frac{\partial}{\partial t} u_f(x,t) = \frac{1}{2} \frac{\partial^2}{\partial x^2} u_f(x,t) - \psi( u_f(x,t)), \quad x \in {\mathbb R},\;t>0 \tag{3}$ with initial condition $$u_f(x,0) = f(x)$$. Since the analogous object to (3) for branching Brownian motions is called the Fisher-Kolmogorov-Petrovski-Piscounov (FKPP) equation, in this paper, the authors call (3) the FKPP equation for $$\psi$$-super-Brownian motions as well. By virtue of the general theory, the $$\psi$$-super-Brownian motions can be categorised into the supercritical, critical and subcritical ones, which just correspond, respectively, to the cases of $$\psi'(0+) < 0$$, $$\psi'(0+) = 0$$ and $$\psi'(0+) > 0$$. The class of $$\psi$$-super-Brownian motions treated here are assumed to be supercritical. Such processes may exhibit an explosive behaviour, however, under the conditions assumed above, $$X$$ remains possibly finite at all positive times. Let us consider the probability of the event $${\mathcal E}=\{ \lim_{t \uparrow \infty} \| X_t \| = 0 \}$$, and assume that $\int^{\infty} \frac{1}{ \sqrt{ \int_{\lambda^*}^{\xi} \psi(u) du } } d \xi < \infty, \tag{4}$ where $$\lambda^*$$ is the largest root of the equation $$\psi(\lambda) =0$$. Non-increasing solutions to (3) of the form $$\Phi_c( x - ct)$$ are specifically interesting objects, where $$\Phi_c \geq 0$$ and $$c$$ is the wave speed. Actually, $$\Phi_c$$ solves $$\frac{1}{2} \Phi_c''+c \cdot \Phi_c'-\psi( \Phi_c)=0$$. For convenience, we write $$\underline{\lambda}=\sqrt{ - 2 \psi'(0+)}$$, and, for each $$\lambda \in {\mathbb R}$$, define $$c_{\lambda}=- \psi'(0+)/ \lambda + \lambda/2$$. Note that, for $$\lambda \in (0, \underline{\lambda}]$$, $$c_{\lambda}$$ has range $$[ \underline{\lambda}, \infty)$$. To analyze the situation precisely, the authors introduce two families of $$\operatorname{P}$$-martingales with respect to the natural filtration $${\mathcal F}_t=\sigma( X_u; u \leqslant t )$$. For $$\lambda \in {\mathbb R}$$, the process $$W_t(\lambda)=e^{- \lambda c_{\lambda} t} \langle e^{- \lambda(\cdot)}, X_t(\cdot) \rangle$$, $$t \geq 0$$, is a martingale, which is a non-negative martingale and, therefore, converges almost surely. For $$\lambda \in {\mathbb R}$$, the process $$\partial W_t(\lambda)=- \frac{\partial}{\partial \lambda} W_t(\lambda)=\langle ( \lambda t + ( \cdot)) e^{- \lambda ( c_{\lambda}t + ( \cdot))}$$, $$X_t \rangle$$, $$t \geq 0$$, is also a martingale, which produces a signed martingale which does not necessarily converges almost surely. Here, are the main results in this paper.
Theorem A.
(i)
The almost sure limit of $$W(\lambda)$$, denoted by $$W_{\infty}(\lambda)$$, is an $$L^1(\operatorname{P})$$-limit if and only if $$| \lambda | \leqslant \underline{\lambda}$$ and $$\int_{[1, \infty)} r ( \log r) \nu(d r) < \infty$$. When $$W_{\infty}(\lambda)$$ is an $$L^1(\operatorname{P})$$-limit, the event $$\{ W_{\infty}(\lambda) > 0 \}$$ agrees with $${\mathcal E}^c$$, $$\operatorname{P}$$-almost surely. Otherwise, when it is not an $$L^1(\operatorname{P})$$-limit, its limit is identically zero.
(ii)
Assume that (4) holds. The martingale $$\partial W( \lambda)$$ has an almost sure non-negative limit when $$| \lambda | \geq \underline{\lambda}$$, which is identically zero when $$| \lambda | > \underline{\lambda}$$, and when $$| \lambda | = \underline{\lambda}$$ its limit is almost surely strictly positive on $${\mathcal E}^c$$ if and only if $$\int_{[1, \infty)} r ( \log r)^2 \nu(dr)<\infty$$.
Theorem B. Assume (4).
(i)
Suppose that $$\int_{[1, \infty)} r (\log r) \nu(dr) < \infty$$ and $$\lambda \in( 0, \underline{\lambda})$$. Then, up to an additive constant in its argument, the travelling wave solution $$\Phi_{c_{\lambda}}$$ is given by $\Phi_{ c_{\lambda}}(x) = - \log \operatorname{E} [ e^{- e^{- \lambda x}} W_{\infty}(\lambda) ],$ and there is a constant $$k_{\lambda} \in (0, \infty)$$ such that $$\lim_{ x \to \infty} \Phi_{ c_{\lambda}}(x)/ e^{- \lambda x} = k_{\lambda}.$$
(ii)
Suppose that $$\int_{[1, \infty)} r ( \log r)^2 \nu(dr) < \infty$$ and $$\lambda = \underline{\lambda}$$. Then, the critical travelling wave solution $$\Phi_{\underline{\lambda}}$$ is given by $\Phi_{\underline{\lambda}}(x) = - \log \operatorname{E}[ e^{- e^{- \underline{\lambda}x} \partial W_{\infty}(\lambda)}].$ Moreover, there is a constant $$k_{\lambda} \in (0, \infty)$$ such that $$\lim_{x \to \infty} \Phi_{c_{\lambda}}(x) /(x e^{- \lambda x} ) = k_{\lambda}.$$

##### MSC:
 60J68 Superprocesses 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60E10 Characteristic functions; other transforms
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