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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.
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.
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).
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 \( \lim_{ x \to \infty} \Phi_{ c_{\lambda}}(x)/ e^{- \lambda x} = k_{\lambda}. \)
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 \( \lim_{x \to \infty} \Phi_{c_{\lambda}}(x) /(x e^{- \lambda x} ) = k_{\lambda}. \)

60J68 Superprocesses
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60E10 Characteristic functions; other transforms
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[1] D. G. Aronson and H. F. Weinberger. Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30 (1978) 33-76. · Zbl 0407.92014 · doi:10.1016/0001-8708(78)90130-5
[2] J. D. Biggins and A. E. Kyprianou. Measure change in multitype branching. Adv. in Appl. Probab. 36 (2004) 544-581. · Zbl 1056.60082 · doi:10.1239/aap/1086957585
[3] M. Bramson. Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Amer. Math. Soc. 44 (1983) iv \(+\) 190 pp. · Zbl 0517.60083
[4] B. Chauvin. Multiplicative martingales and stopping lines for branching Brownian motion. Ann. Probab. 30 (1991) 1195-1205. · Zbl 0738.60079 · doi:10.1214/aop/1176990340
[5] R. Durrett. Probability Theory and Examples , 2nd edition. Duxbury Press, 1996. · Zbl 1202.60001
[6] R. Durrett and C. Neuhauser. Particle systems and reaction-diffusion equations. Ann. Probab. 22 (1994) 289-333. · Zbl 0799.60093 · doi:10.1214/aop/1176988861
[7] E. B. Dynkin. A probabilistic approach to one class of non-linear differential equations. Probab. Theory Related Fields 89 (1991) 89-115. · Zbl 0722.60062 · doi:10.1007/BF01225827
[8] E. B. Dynkin. Branching particle systems and superprocesses. Ann. Probab. 19 (1991) 1157-1194. · Zbl 0732.60092 · doi:10.1214/aop/1176990339
[9] E. B. Dynkin. Superprocesses and partial differential equations. Ann. Probab. 21 (1993) 1185-1262. · Zbl 0806.60066 · doi:10.1214/aop/1176989116
[10] E. B. Dynkin. Branching exit Markov systems and superprocesses. Ann. Probab. 29 (2001) 1833-1858. · Zbl 1014.60079 · doi:10.1214/aop/1015345774
[11] E. B. Dynkin. Diffusions, Superdiffusions and Partial Differential Equations . Amer. Math. Soc., Providence, RI, 2002. · Zbl 0999.60003
[12] E. B. Dynkin and S. E. Kuznetsov. \(\mathbb{N}\)-measures for branching Markov exit systems and their applications to differential equations. Probab. Theory Related Fields 130 (2004) 135-150. · Zbl 1068.31002 · doi:10.1007/s00440-003-0333-8
[13] J. Engländer and A. E. Kyprianou. Local extinction versus local exponential growth for spatial branching processes. Ann. Probab. 32 (2004) 78-99. · Zbl 1056.60083 · doi:10.1214/aop/1078415829
[14] S. N. Evans. Two representations of a conditioned superprocess. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 959-971. · Zbl 0784.60052 · doi:10.1017/S0308210500029619
[15] P. C. Fife and J. B. McLeod. The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Ration. Mech. Anal. 65 (1977) 335-361. · Zbl 0361.35035 · doi:10.1007/BF00250432
[16] R. A. Fisher. The wave of advance of advantageous genes. Ann. Eugenics 7 (1937) 355-369. · JFM 63.1111.04
[17] P. J. Fitzsimmons. Construction and regularity of measure-valued Markov branching processes. Israel J. Math. 64 (1988) 337-361. · Zbl 0673.60089 · doi:10.1007/BF02882426
[18] Y. Git, J. W. Harris and S. C. Harris. Exponential growth rates in a typed branching diffusion. Ann. Appl. Probab. 17 (2007) 609-653. · Zbl 1131.60077 · doi:10.1214/105051606000000853
[19] D. R. Grey. Asymptotic behavior of continuous time, continuous state-space branching processes. J. Appl. Probab. 11 (1974) 669-677. · Zbl 0301.60060 · doi:10.2307/3212550
[20] R. Hardy and S. C. Harris. A spine approach to branching diffusions with applications to \(L^{p}\)-convergence of martingales. In Séminaire de Probabilités XLII 281-330. Berlin, 2009. · Zbl 1193.60100 · doi:10.1007/978-3-642-01763-6_11
[21] S. C. Harris. Travelling waves for the F-K-P-P equation via probabilistic arguments. Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 503-517. · Zbl 0946.35040 · doi:10.1017/S030821050002148X
[22] S. C. Harris and M. Roberts. Measure changes with extinction. Stat. Probab. Lett. 79 (2009) 1129-1133. · Zbl 1163.60309 · doi:10.1016/j.spl.2008.12.025
[23] Y. Kametaka. On the nonlinear diffusion equation of Kolmogorov-Petrovskii-Piskunov type. Osaka J. Math. 13 (1976) 11-66. · Zbl 0344.35050
[24] A. Kolmogorov, I. Petrovskii and N. Piskounov. Étude de l’équation de la diffusion avec croissance de la quantité de la matière at son application a un problèm biologique. Moscow Univ. Math. Bull. 1 (1937) 1-25. · Zbl 0018.32106
[25] A. E. Kyprianou. Travelling wave solution to the K-P-P equation: Alternatives to Simon Harris’ probabilistic analysis Ann. Inst. Henri Poincaré Probab. Stat. 40 (2004) 53-72. · Zbl 1042.60057 · doi:10.1016/S0246-0203(03)00055-4 · numdam:AIHPB_2004__40_1_53_0 · eudml:77799
[26] A. E. Kyprianou. Asymptotic radial speed of the support of supercritical branching Brownian motion and super-Brownian motion in \(\mathbb{R}^{d}\). Markov Process. Related Fields 11 (2005) 145-156. · Zbl 1076.60074
[27] A. E. Kyprianou. Introductory Lectures on Fluctuations of Lévy Processes with Applications . Springer, Berlin, 2006. · Zbl 1104.60001
[28] A. E. Kyprianou and A. Murillo-Salas. Super-Brownian motion: \(L^{p}\)-convergence of martingales through the pathwise spine decomposition. Preprint, 2011. Available at . · Zbl 1279.60111 · arxiv.org
[29] K.-S. Lau. On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov. J. Differential Equations 59 (1985) 44-70. · Zbl 0584.35091 · doi:10.1016/0022-0396(85)90137-8
[30] J. F. Le Gall. Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics ETH Zürich . Birkhäuser, Basel, 1999. · Zbl 0938.60003
[31] R.-L. Liu, Y.-X. Ren and R. Song. \(L\log L\) criterion for a class of superdiffusions. J. Appl. Probab. 46 (2009) 479-496. · Zbl 1175.60077 · doi:10.1239/jap/1245676101
[32] R. Lyons. A simple path to Biggins’ martingale convergence theorem. In Classical and Modern Branching Processes 217-222. K. B. Athreya and P. Jagers (Eds). Springer, New York, 1997. · Zbl 0897.60086 · doi:10.1007/978-1-4612-1862-3_17
[33] R. Lyons, R. Pemantle and Y. Peres. Conceptual proofs of \(L\log L\) criteria for mean behaviour of branching processes. Ann. Probab. 23 (1995) 1125-1138. · Zbl 0840.60077 · doi:10.1214/aop/1176988176
[34] P. Maillard. The number of absorbed individuals in branching Brownian motion with a barrier. Available at . 1004.1426 · Zbl 1281.60070 · arxiv.org
[35] H. P. McKean. Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Comm. Pure Appl. Math. 28 (1975) 323-331. · Zbl 0316.35053 · doi:10.1002/cpa.3160280302
[36] J. Neveu. Multiplicative martingales for spatial branching processes. In Seminar on Stochastic Processes 1987 223-242. E. Çinlar, K. L. Chung and R. K. Getoor (Eds). Progress in Probability and Statistics 15 . Birkhäuser, Boston, 1988. · Zbl 0652.60089
[37] R. G. Pinsky. K-P-P-type asymptotics for nonlinear diffusion in a large ball with infinite boundary data and on \(\mathbf{R}^d\) with infinite initial data outside a large ball. Comm. Partial Differential Equations 20 (1995) 1369-1393. · Zbl 0831.35078 · doi:10.1080/03605309508821136
[38] Y.-X. Ren and H. Wang. On states of total weighted occupation times of a class of infinitely divisible superprocesses on a bounded domain. Potential Anal. 28 (2008) 105-137. · Zbl 1205.60151 · doi:10.1007/s11118-007-9073-1
[39] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften 293 . Springer, Berlin, 1980.
[40] Y. C. Sheu. Lifetime and compactness of range for \(\psi\)-super-Brownian motion with a general branching mechanism. Stochastics Process. Appl. 70 (1997) 129-141. · Zbl 0911.60060 · doi:10.1016/S0304-4149(97)00059-8
[41] K. Uchiyama. The behavior of solutions of some non-linear diffusion equations for large time. J. Math. Kyoto Univ. 18 (1978) 453-508. · Zbl 0408.35053
[42] A. I. Volpert, V. A. Volpert and V. A. Volpert. Traveling Wave Solutions of Parabolic Systems. Translations of Mathematical Monographs 140 . Amer. Math. Soc., 1994. · Zbl 1017.34014 · doi:10.1016/S0362-546X(01)00103-1
[43] S. Watanabe. A limit theorem of branching processes and continuous-state branching processes. J. Math. Kyoto Univ. 8 (1968) 141-167. · Zbl 0159.46201
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