LAUSR

In this paper, we consider the stochastic heat equation of the form ∂u∂t=Δαu+∂2B∂t∂x,$$\frac{\partial u}{\partial t}=\Delta_{\alpha}u+\frac{\partial ^{2}B}{\partial t\,\partial x}, $$ where ∂2B∂t∂x$\frac{\partial^{2}B}{\partial t\,\partial x}$ is a fractional Brownian sheet with Hurst indices H1,H2∈(12,1)$H_{1},H_{2}\in(\frac{1}{2},1)$ and Δα=−(−Δ)α/2$\Delta _{\alpha}=-(-\Delta)^{\alpha/2}$ is a fractional Laplacian operator with 1<α≤2$1<\alpha\leq2$. In particular, when H2=12$H_{2}=\frac{1}{2}$ we show that the temporal process {u(t,⋅),0≤t≤T}$\{u(t,\cdot),0\leq t\leq T\}$ admits a nontrivial p-variation with p=2α2αH1−1$p=\frac{2\alpha}{2\alpha H_{1}-1}$ and study its local nondeterminism and existence of the local time.