# American Institute of Mathematical Sciences

August  2013, 18(6): 1683-1696. doi: 10.3934/dcdsb.2013.18.1683

## Exponential growth rate for a singular linear stochastic delay differential equation

 1 Institut für Mathematik, Technische Universität Berlin, Str. des 17. Juni 136, 10623 Berlin

Received  January 2012 Revised  March 2012 Published  March 2013

We establish the existence of a deterministic exponential growth rate for the norm (on an appropriate function space) of the solution of the linear scalar stochastic delay equation $d X(t) = X(t-1) d W(t)$ which does not depend on the initial condition as long as it is not identically zero. Due to the singular nature of the equation this property does not follow from available results on stochastic delay differential equations. The key technique is to establish existence and uniqueness of an invariant measure of the projection of the solution onto the unit sphere in the chosen function space via asymptotic coupling and to prove a Furstenberg-Hasminskii-type formula (like in the finite dimensional case).
Citation: Michael Scheutzow. Exponential growth rate for a singular linear stochastic delay differential equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1683-1696. doi: 10.3934/dcdsb.2013.18.1683
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