Some properties of the solution to fractional heat equation with a fractional Brownian noise

In this paper, we consider the stochastic heat equation of the form $$\frac{\partial u}{\partial t}=\Delta_{\alpha}u+\frac{\partial ^{2}B}{\partial t\,\partial x}, $$ where $\frac{\partial^{2}B}{\partial t\,\partial x}$ is a fractional Brownian sheet with Hurst indices $H_{1},H_{2}\in(\frac{1}{2},1)$ and $\Delta _{\alpha}=-(-\Delta)^{\alpha/2}$ is a fractional Laplacian operator with $1<\alpha\leq2$ . In particular, when $H_{2}=\frac{1}{2}$ we show that the temporal process $\{u(t,\cdot),0\leq t\leq T\}$ admits a nontrivial p-variation with $p=\frac{2\alpha}{2\alpha H_{1}-1}$ and study its local nondeterminism and existence of the local time.

Abstract licence: CC BY

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