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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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