Stochastic Heat Equation with Multiplicative Fractional-Colored Noise
By R. Balan, Ciprian A. Tudor. 2008
55 citations recorded by the source corpus.
AbstractWe consider the stochastic heat equation with multiplicative noise $u_{t}=\frac{1}{2}\Delta u+u\dot{W}$ in ℝ+×ℝd, whose solution is interpreted in the mild sense. The noise $\dot{W}$ is fractional in time (with Hurst index H≥1/2), and colored in space (with spatial covariance kernel f). When H>1/2, the equation generalizes the Itô-sense equation for H=1/2. We prove that if f is the Riesz kernel of order α, or the Bessel kernel of order α1/2), respectively d<2+α (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 kth 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.
Subfields: Control and Systems Engineering, Finance
View Paper: https://www.semanticscholar.org/paper/5b3978b94a88963e62729d6f3941039ac59dffe3
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Citations in the corpus
- Malliavin Calculus for Fractional Delay Equations (2009, background): “Just to mention a few examples concerning fractional Brownian motion with Hurst parameter H > 1/2, let us quote ordinary differential equations [18, 30, 31, 38], some interesting cases of PDEs [3, 7, 10, 11, 15, 16, 24, 34], as well as Volterra type systems [8, 9].”