Invariant measure for the stochastic Cauchy problem driven by a cylindrical Lévy process

TY - JOUR

T1 - Invariant measure for the stochastic Cauchy problem driven by a cylindrical Lévy process

AU - Riedle, Markus

AU - Umesh, Umesh

PY - 2020/8/20

Y1 - 2020/8/20

N2 - In this work, we present sufficient conditions for the existence of a stationary solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Lévy process, and show that these conditions are also necessary if the semigroup is stable, in which case the invariant measure is unique. For typical situations such as the heat equation, we significantly simplify these conditions without assuming any further restrictions on the driving cylindrical Lévy process and demonstrate their application in some examples.

AB - In this work, we present sufficient conditions for the existence of a stationary solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Lévy process, and show that these conditions are also necessary if the semigroup is stable, in which case the invariant measure is unique. For typical situations such as the heat equation, we significantly simplify these conditions without assuming any further restrictions on the driving cylindrical Lévy process and demonstrate their application in some examples.

KW - cylindrical Lévy processes, Cauchy problem, invariant measures, stationary distributions, Mehler semigroup

UR - https://arxiv.org/abs/1904.03118

M3 - Article

SN - 0022-247X

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

ER -