Abstract
We study the asymptotic behavior as $t \to \infty $ of a time-dependent family $(\mu _{t})_{t \geq 0}$ of probability measures on $\mathbb {R}$ solving the kinetic-type evolution equation $\partial _{t} \mu _{t} + \mu _{t} = Q(\mu _{t})$ where $Q$ is a smoothing transformation on $\mathbb {R}$. This problem has been investigated earlier, e.g. by Bassetti and Ladelli [Ann. Appl. Probab. 22(5): 1928–1961, 2012] and Bogus, Buraczewski and Marynych [Stochastic Process. Appl. 130(2):677–693, 2020]. Combining the refined analysis of the latter paper, which provides a probabilistic description of the solution $\mu _{t}$ as the law of a suitable random sum related to a continuous-time branching random walk at time $t$, with recent advances in the analysis of the extremal positions in the branching random walk we are able to solve the case that has been left open until now. In the course of our work, we significantly weaken the assumptions in the literature that guarantee the existence (and uniqueness) of a solution to the evolution equation $\partial _{t} \mu _{t} + \mu _{t} = Q(\mu _{t})$.
Highlights
Given a sequence A = (A1, A2, . . .) of non-negative random variables with N := max{j : Aj = 0} < ∞ almost surely we consider the kinetic-type evolution equation∂tμt + μt = Q(μt) (1.1)for a time-dependent familyt≥0 of probability measures on R equipped with the Borel σ-algebra B(R) where (1.1) has to be understood in the weak sense and Q is theSelf-similar solutions to kinetic-type evolution equations smoothing transformation associated with A
The smoothing transformation Q is a self-map of M1(R), the set of probability measures on (R, B(R)), defined by the formula
Where φ0 is the Fourier transform of a given μ0 ∈ M1(R) and Q is a self-map of the set of characteristic functions of probability measures on (R, B(R)) defined by
Summary
Self-similar solutions to kinetic-type evolution equations smoothing transformation associated with A. The smoothing transformation Q is a self-map of M1(R), the set of probability measures on (R, B(R)), defined by the formula. Aj Xj , j=1 where L(Y ) denotes the law of a random variable Y and X1, X2, . Where φ0 is the Fourier transform of a given μ0 ∈ M1(R) and Q is a self-map of the set of characteristic functions of probability measures on (R, B(R)) defined by. For φ being the Fourier transform of some probability measure μ ∈ M1(R). Under suitable assumptions (see e.g. Theorem 1.2 below or [12, Proposition 2.5]), given an initial law μ0, Eq (1.1) has a unique solution, which we shall denote by (μt)t≥0 . The corresponding family of Fourier transforms will be denoted by (φt)t≥0.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
