论文标题
动力学碰撞的动力学方程的溶液
Solutions of kinetic-type equations with perturbed collisions
论文作者
论文摘要
我们研究一类运动型微分方程$ \ partial ϕ_t/\ partial t+ϕ_t = \ wideHat {\ MathCal {q}} ϕ_t $，其中$ \ wideHat {\ natercal {\ nathcal {q}} $ nis $ spe n $ se $ se $ geq n $ ge qe qe qe qe qe qe qe qe qe qe qe qe qe qe qe qe qe qe qe q $ ge qe qe qe qe qe qe qe q ege n $ ge qe qe qe qe qe qe qe qe q $ geq n $，傅立叶 - 风格概率措施的变换。我们表明，在$ \ wideHat {\ mathcal {q}} $上的轻度假设下，上面的微分方程具有独特的解决方案，并表示该解决方案是与连续分支随机步行相关的特定随机过程的特征功能，与$ \ wideHat {\ wideHat {\ nathcal {q}} $有关。为此过程建立限制定理，使我们可以将解决方案的渐近性特性描述为$ t \ to \ infty $。
We study a class of kinetic-type differential equations $\partial ϕ_t/\partial t+ϕ_t=\widehat{\mathcal{Q}}ϕ_t$, where $\widehat{\mathcal{Q}}$ is an inhomogeneous smoothing transform and, for every $t\geq 0$, $ϕ_t$ is the Fourier--Stieltjes transform of a probability measure. We show that under mild assumptions on $\widehat{\mathcal{Q}}$ the above differential equation possesses a unique solution and represent this solution as the characteristic function of a certain stochastic process associated with the continuous time branching random walk pertaining to $\widehat{\mathcal{Q}}$. Establishing limit theorems for this process allows us to describe asymptotic properties of the solution, as $t\to\infty$.
