Infinite rate symbiotic branching on the real line: the tired frogs model. (English. French summary) Zbl 1465.60085

This paper treats a special case of the infinite rate symbiotic branching process on the real line with different motion speeds for the two types, i.e., the tired frogs model as the authors call it. The modeling interpretation behind the setting is quite stimulating and interesting. Initially the frogs on the positive half-line are dormant while those on the negative half-line are awake and move according to the heat flow. At the interface, the incoming wake frogs try to wake up the dormant frogs and succeed with a probability proportional to their amount among the total amount of involved frogs at the specific site. Otherwise, the incoming frogs also fall asleep. By regarding this frog model as the limit of approximating processes and computing the structure of jumps, the authors show that their frog model can be described by a stochastic partial differential equation on the real line with a jump-type noise.
More precisely, let \(M_F\) be the space of finite measures on \({\mathbb R}\) equipped with the weak topology, and \(D_U\) is the space of cadlag functions \([0, \infty) \to U\) equipped with the Skorokhod topology for any metric space \(U\). \(C_b^2({\mathbb R})\) denotes the space of twice continuously differentiable functions with bounded first and second derivative. Let \(X_t^1\) be a process satisfying \[\partial_t X_t^1 = \frac{1}{2} \partial_x^2 X_t^1 \qquad \text{on} \quad (- \infty, z_n) \quad \text{for} \quad \tau^n < t <\tau^{n+1} \] with \(X_t^1(x) =0\) for all \(x \geq \bar{\ell}_t\), where \(0 < z_1 < z_2 < \cdots\). \(( X_t^1, X_t^2)\), \(t \geq 0\), are \(M_F\)-valued processes, and initially \(X_0^2 = \sum_{i \geq 1} x_i \delta_{ z_i}\) with \[\sum_i x_i < \infty, \qquad \text{and} \qquad \bar{\ell}_t := \sup \{ x : \, X_t^2 ( ( - \infty, x ]) = 0 \}. \] Define \(\tau^i := \inf \{ t \geq 0: \, \bar{\ell}_t = z_i \}\). \(X_t^2\) satisfies \[ X_t^2( \{ z_n \} ) = X_0^2 ( \{ z_n \} ) + \int_{\tau^n}^t \frac{1}{2} \partial_x^2 X_s^1 ( \{ z_n \}) ds \qquad \text{for} \quad \tau^n < t < \tau^{n+1}. \] Assume that (i) \(X_t^1\) is an absolutely continuous finite measure with compact support in \((- \infty, 0]\) and with bounded density \(X_0^1(x)\); (ii) \(X_0^2\) is an absolutely continuous finite measure with support in \([0,1]\) and with density \(X_0^2(x)\); (iii) \(X_0^2(x) > 0\) for all \(x \in (0,1)\) and \(X_0^2(x)\) is continuous in \(x \in (0,1)\). The family of approximating processes \(( X^{1, \eta}, X^{2, \eta})\) are defined as follows. For any \(\eta > 0\), define \[ X_0^{1, \eta} = X_0^1, \qquad x_i^{2,\eta} = X_0^2 ( ( (i-1)\eta, i \eta] ) \quad \text{for} \quad i \geq 1, \qquad X_0^{2, \eta} = \sum_{i \geq 1} x_i^{2, \eta} \delta_{i \eta}. \] As a matter of fact, they show that \(( X^{1, \eta}, X^{2, \eta})\), \(\eta > 0\), is tight in \(D_{M_F \times M_F}\) and prove that any limit point for \(\eta \searrow 0+\) is a weak solution to the following system of stochastic partial differential equations: for any \(\phi_1, \phi_2 \in C_b^2({\mathbb R})\), \begin{align*} &X_t^1(\phi_1) = X_0^1(\phi_1) + \int_0^t X_s^1 ( \frac{1}{2} \phi_1'') ds + M_t(\phi_1), \\ &X_t^2(\phi_2) = X_0^2(\phi_2) - M_t(\phi_2), \end{align*} where \(M_t(\phi_i)\), \(i=1,2\), are martingales derived from the orthogonal martingale measure \({\mathcal M}\) by \[ M_t(\phi) := \int_-^t \int_0^{\infty} Y_{s-} \phi( \ell_{s-} ) I_{ [0, i(X_{s-})]}(a) {\mathcal M}(ds, da), \] \(\ell_s := \inf \{ x : \, X_s^2( ( - \infty, x]) > 0 \} \wedge 1\) anf \(Y_s = X_s^2( \{ \ell_s \} )\), where \({\mathcal N}(dt, dr)\) is a Poisson point process on \({\mathbb R}_+^2\) with intensity measure \({\mathcal N}'(dt, dr)\) \(=\) \(dt dr\), and \({\mathcal M}\) \(:=\) \({\mathcal N} - {\mathcal N}'\).
For other related works, see, e.g., [J. Blath et al., Ann. Probab. 44, No. 2, 807–866 (2016; Zbl 1347.60119)] for a symbiotic branching model; [A. Klenke and L. Mytnik, Probab. Theory Relat. Fields 154, No. 3–4, 533–584 (2012; Zbl 1266.60160)] for infinite rate mutually catalytic branching in infinitely many colonies; see also [the authors, Ann. Probab. 40, No. 1, 103–129 (2012; Zbl 1244.60088)] for the longtime behaviors of the same model as the above.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J68 Superprocesses
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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