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Kamil Bogus

Publications and source records attributed to Kamil Bogus.

5 recordsLinked to original sources
arXiv2019

Self-similar solutions of kinetic-type equations: the boundary case

For a time dependent family of probability measures $(ρ_t)_{t\ge 0}$ we consider a kinetic-type evolution equation $\partial ϕ_t/\partial t + ϕ_t = \widehat{Q} ϕ_t$ where $\widehat{Q}$ is a smoothing transform and $ϕ_t$ is the Fourier--Stieltjes transform of $ρ_t$. Assuming that the initial measure $ρ_0$ belongs to the domain of attraction of a stable law, we describe asymptotic properties of $ρ_t$, as $t\to\infty$. We consider the critical regime when the standard normalization leads to a degenerate limit and find an appropriate scaling ensuring a non-degenerate self-similar limit. Our approach is based on a probabilistic representation of probability measures $(ρ_t)_{t\ge 0}$ that refines the corresponding construction proposed in Bassetti and Ladelli [Ann. Appl. Probab. 22(5): 1928--1961, 2012].

math.PR↗
arXiv2017

Asymptotic behaviour of the Bessel heat kernels

We consider Dirichlet heat kernel $p_a^{(μ)}(t,x,y)$ for the Bessel differential operator $L^{(μ)}=\frac{d^2}{dx^2}+\frac{2μ+1}{2x}$, $μ\in\mathbb{R}$, in half-line $(a,\infty)$, $a>0$, and provide its asymptotic expansions for $xy/t\rightarrow\infty$.

math.AP↗
arXiv2015

Sharp estimates of Green function of hyperbolic Brownian Motion

The main objective of the work is to provide sharp two-sided estimates of $λ$-Green function of hyperbolic Brownian motion of a half-space. We strongly rely on recent results obtained by K. Bogus and J. Malecki [3], regarding precise estimates of the Bessel heat kernel of half-lines.

math.PR↗
arXiv2015

Heat kernel estimates for the Bessel differential operator in half-line

In the paper we consider the Bessel differential operator L^(μ)=\dfrac{d^2}{dx^2}+\dfrac{2μ+1}{x}\dfrac{d}{dx} in half-line (a,\infty), a>0, and its Dirichlet heat kernel p_a^(μ)(t,x,y). For μ=0, by combining analytical and probabilistic methods, we provide sharp two-sided estimates of the heat kernel for the whole range of the space parameters x,y>a and every t>0, which complements the recent results given in [1], where the case μ\neq 0 was considered.

math.AP↗