Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

Raluca Balan (1) , Ciprian A. Tudor (2)

1 Department of Mathematics and Statistics [Ottawa]
2 SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne)

We consider the stochastic heat equation with multiplicative noise $u_t=\frac{1}{2}\Delta u+ u \diamond \dot{W}$ in $\bR_{+} \times \bR^d$, where $\diamond$ denotes the Wick product, and the solution is interpreted in the mild sense. The noise $\dot W$ is fractional in time (with Hurst index $H \geq 1/2$), and colored in space (with spatial covariance kernel $f$). We prove that if $f$ is the Riesz kernel of order $\alpha$, or the Bessel kernel of order $\alpha1/2$), respectively $d<2+\alpha$ (if $H=1/2$), whereas if $f$ is the heat kernel or the Poisson kernel, then the equation has a solution for any $d$. We give a representation of the $k$-th order moment of the solution, in terms of an exponential moment of the ``convoluted weighted'' intersection local time of $k$ independent $d$-dimensional Brownian motions.

Domaines: Probabilités [math.PR]

Raluca Balan, Ciprian A. Tudor. Stochastic Heat Equation with Multiplicative Fractional-Colored Noise. Journal of Theoretical Probability, 2010, 23 (3), pp.834-870. ⟨10.1007/s10959-009-0237-3⟩. ⟨hal-00345923v2⟩