Page 673 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 673
PROBABILITY
[2] S. HERRMANN AND C. ZUCCA, Exact simulation of the first-passage time of diffusions,
Journal of Scientific Computing, 2019.
The first exit problem of reaction-diffusion equations for small
multiplicative Lévy noise
Michael A. Hoegele, ma.hoegele@uniandes.edu.co
Universidad de los Andes, Colombia
In this talk we present the first exit problem from the vicinity of a stable state in a generic class
of dissipative, scalar reaction diffusion equations perturbed by multiplicative, regularly varying
Lévy noise in the limit of small noise intensity. In finite dimensions the asymptotic behavior of
the first exit time and location have been determined in recent years by Pavylykevich, Imkeller,
and H. and in infinite dimensions the additive case had been studied for the Chafee-Infante
equation by Debussche, H. and Imkeller. We show how to obtain these results for perturbations
by multiplicative regularly varying noise. For this aim we use some exponential estimates in
infinite dimensions of the stochastic convolution with Lévy processes with bounded jump size.
The main example covered by our results are the linear heat equation and multi-well gradient
equations, such as the Chafee-Infante equation, perturbed by additive and multiplicative α-
stable noise.
Joint functional convergence of partial sum and maxima processes
Danijel Krizmanic´, dkrizmanic@math.uniri.hr
University of Rijeka, Croatia
For a strictly stationary sequence of random variables we study functional convergence of the
joint partial sum and partial maxima process under joint regular variation with index α ∈ (0, 2)
and weak dependence conditions. The convergence takes place in the space of R2–valued càdlàg
functions on [0, 1] with the Skorohod weak M1 topology, and the limiting process consists of
an α–stable Lévy process and an extremal process. We also show that the weak M1 topology in
general can not be replaced by the standard M1 topology.
Markov chains in stationary and ergodic random environment
Attila Lovas, lovas.attila@renyi.hu
Alfréd Rényi Institute of Mathematics, Hungary
Coauthor: Miklós Rásonyi
Markov chains in stationary random environments (MCREs) with a general (not necessarily
countable) state-space appear in several branches of applied probability including mathematical
finance, queuing models with non-i.i.d. service times and statistical learning theory. Assuming
suitable versions of the standard drift and minorization conditions, we prove the existence of
limiting distributions for MCREs in cases when the system dynamics is contractive on the aver-
age with respect to the Lyapunov function and large enough small sets exist with large enough
minorization constants. We also establish that a law of large numbers holds for bounded func-
671
[2] S. HERRMANN AND C. ZUCCA, Exact simulation of the first-passage time of diffusions,
Journal of Scientific Computing, 2019.
The first exit problem of reaction-diffusion equations for small
multiplicative Lévy noise
Michael A. Hoegele, ma.hoegele@uniandes.edu.co
Universidad de los Andes, Colombia
In this talk we present the first exit problem from the vicinity of a stable state in a generic class
of dissipative, scalar reaction diffusion equations perturbed by multiplicative, regularly varying
Lévy noise in the limit of small noise intensity. In finite dimensions the asymptotic behavior of
the first exit time and location have been determined in recent years by Pavylykevich, Imkeller,
and H. and in infinite dimensions the additive case had been studied for the Chafee-Infante
equation by Debussche, H. and Imkeller. We show how to obtain these results for perturbations
by multiplicative regularly varying noise. For this aim we use some exponential estimates in
infinite dimensions of the stochastic convolution with Lévy processes with bounded jump size.
The main example covered by our results are the linear heat equation and multi-well gradient
equations, such as the Chafee-Infante equation, perturbed by additive and multiplicative α-
stable noise.
Joint functional convergence of partial sum and maxima processes
Danijel Krizmanic´, dkrizmanic@math.uniri.hr
University of Rijeka, Croatia
For a strictly stationary sequence of random variables we study functional convergence of the
joint partial sum and partial maxima process under joint regular variation with index α ∈ (0, 2)
and weak dependence conditions. The convergence takes place in the space of R2–valued càdlàg
functions on [0, 1] with the Skorohod weak M1 topology, and the limiting process consists of
an α–stable Lévy process and an extremal process. We also show that the weak M1 topology in
general can not be replaced by the standard M1 topology.
Markov chains in stationary and ergodic random environment
Attila Lovas, lovas.attila@renyi.hu
Alfréd Rényi Institute of Mathematics, Hungary
Coauthor: Miklós Rásonyi
Markov chains in stationary random environments (MCREs) with a general (not necessarily
countable) state-space appear in several branches of applied probability including mathematical
finance, queuing models with non-i.i.d. service times and statistical learning theory. Assuming
suitable versions of the standard drift and minorization conditions, we prove the existence of
limiting distributions for MCREs in cases when the system dynamics is contractive on the aver-
age with respect to the Lyapunov function and large enough small sets exist with large enough
minorization constants. We also establish that a law of large numbers holds for bounded func-
671
