Jani Lukkarinen

We provide non-asymptotic error bounds in the path Wasserstein distance with quadratic integral cost between suitable functionals of the telegraph process and the corresponding functional of Brownian motion with explicit diffusivity constant. These results cover, in particular, the well-known example of the exponential integral functional of the geometric Brownian motion. The non-asymptotic error bounds tend to zero in the so-called Kac regime. Moreover, the error bounds remain valid when the flip rate for the telegraph process is small. We assess the sharpness of the error bounds through numerical experiments.

It is well known that for a large class of coagulation kernels, Smoluchowski coagulation equations have particular power law solutions which yield a constant flux of mass along all scales of the system. In this paper, we prove that for some choices of the coagulation kernels there are solutions with a constant flux of mass along all scales which are not power laws. The result is proved by means of a bifurcation argument.

We study the long-time behaviour of the solutions to Smoluchowski coagulation equations with a source term of small clusters. The source drives the system out-of-equilibrium, leading to a rich range of different possible long-time behaviours, including anomalous self-similarity. The coagulation kernel is non-gelling, homogeneous, with homogeneity γ ⩽ − 1 , and behaves like x γ + λ y − λ when y ≪ x with γ + 2 λ > 1 . Our analysis shows that the long-time behaviour of the solutions depends on the parameters γ and λ. More precisely, we argue that the long-time behaviour is self-similar, although the scaling of the self-similar solutions depends on the sign of γ + λ and on whether γ = − 1 or γ < − 1 . In all these cases, the scaling differs from the usual one that has been previously obtained when γ + 2 λ < 1 or γ + 2 λ ⩾ 1 , γ > − 1 . In the last part of the paper, we present some conjectures supporting the self-similar ansatz also for the critical case γ + 2 λ = 1 , γ ⩽ − 1 .

The existence and non-existence of stationary solutions of multicomponent coagulation equations with a constant flux of mass towards large sizes is investigated. The flux may be induced by a source of small clusters or by a flux boundary condition at the origin of the composition space, and the coagulation kernel can be very general, merely satisfying certain power law asymptotic bounds in terms of the total number of monomers in a cluster. Our set-up, including an appropriate definition of multicomponent flux, allows a sharp classification of the existence of stationary solutions. In particular, this analysis extends previous results for one-component systems to a larger class of kernels.

It is well known that for a large class of coagulation kernels, Smoluchowski coagulation equations have particular power law solutions which yield a constant flux of mass along all scales of the system. In this paper, we prove that for some choices of the coagulation kernels there are solutions with a constant flux of mass along all scales which are not power laws. The result is proved by means of a bifurcation argument.

In this manuscript, we provide a non-asymptotic process level control between the telegraph process and the Brownian motion with suitable diffusivity constant via a Wasserstein distance with quadratic average cost. In addition, we derive non-asymptotic estimates for the corresponding time average $p$-th moments. The proof relies on coupling techniques such as coin-flip coupling, synchronous coupling and the Koml\'os--Major--Tusn\'ady coupling.

We consider the multicomponent Smoluchowski coagulation equation under non-equilibrium conditions induced either by a source term or via a constant flux constraint. We prove that the corresponding stationary non-equilibrium solutions have a universal localization property. More precisely, we show that these solutions asymptotically localize into a direction determined by the source or by a flux constraint: the ratio between monomers of a given type to the total number of monomers in the cluster becomes ever closer to a predetermined ratio as the cluster size is increased. The assumptions on the coagulation kernel are quite general, with isotropic power law bounds. The proof relies on a particular measure concentration estimate and on the control of asymptotic scaling of the solutions which is allowed by previously derived estimates on the mass current observable of the system.