On a continuous analog of the parabolic Anderson model

Abstract

UDC 519.21; 517.9

We consider a stochastic equation in $\mathbb{R}^{d},$ whose nonlocal part is a convolution operator with nonnegative symbol and the local part is an operator of multiplication by an ergodic field in $\mathbb{R}^{d}.$ We present the upper and lower bounds for its solutions corresponding to the constant initial data and present an example of the field for which these bounds coalesce as $t\rightarrow \infty $ resulting in an asymptotic formula for the logarithm of the solution. We also briefly discuss the spectral aspect of our results.