Łukasz Stȩpień | Searcharxiv

arXiv2021

Efficient approximation of SDEs driven by countably dimensional Wiener process and Poisson random measure

In this paper we deal with pointwise approximation of solutions of stochastic differential equations (SDEs) driven by infinite dimensional Wiener process with additional jumps generated by Poisson random measure. The further investigations contain upper error bounds for the proposed truncated dimension randomized Euler scheme. We also establish matching (up to constants) upper and lower bounds for $\varepsilon$-complexity and show that the defined algorithm is optimal in the Information-Based Complexity (IBC) sense. Finally, results of numerical experiments performed by using GPU architecture are also reported.

math.PR↗

arXiv2012

Long time, large scale limit of the Wigner transform for a system of linear oscillators in one dimension

We consider the long time, large scale behavior of the Wigner transform $W_\eps(t,x,k)$ of the wave function corresponding to a discrete wave equation on a 1-d integer lattice, with a weak multiplicative noise. This model has been introduced in Basile, Bernardin, and Olla to describe a system of interacting linear oscillators with a weak noise that conserves locally the kinetic energy and the momentum. The kinetic limit for the Wigner transform has been shown in Basile, Olla, and Spohn. In the present paper we prove that in the unpinned case there exists $γ_0>0$ such that for any $γ\in(0,γ_0]$ the weak limit of $W_\eps(t/\eps^{3/2γ},x/\eps^γ,k)$, as $\eps\ll1$, satisfies a one dimensional fractional heat equation $\partial_t W(t,x)=-\hat c(-\partial_x^2)^{3/4}W(t,x)$ with $\hat c>0$. In the pinned case an analogous result can be claimed for $W_\eps(t/\eps^{2γ},x/\eps^γ,k)$ but the limit satisfies then the usual heat equation.

math-ph↗