  Let $u(t,x)$ be the solution to a stochastic heat equation $$
\frac{\partial}{\partial t}u=\frac12\frac{\partial^2}{\partial
x^2}u+\frac{\partial^2}{\partial t\partial x}X(t,x),\quad t\geq 0, x\in
{\mathbb R} $$ with initial condition $u(0,x)\equiv 0$, where $X$ is a
time-space white noise. This paper is an attempt to study stochastic analysis
questions of the solution $u(t,x)$. In fact, the solution is a Gaussian process
such that the process $t\mapsto u(t,\cdot)$ is a bi-fractional Brownian motion
seemed a fractional Brownian motion with Hurst index $H=\frac14$ for every real
number $x$. However, the properties of the process $x\mapsto u(\cdot,x)$ are
unknown. In this paper we consider the quadratic covariations of the two
processes $x\mapsto u(\cdot,x),t\mapsto u(t,\cdot)$. We show that $x\mapsto
u(\cdot,x)$ admits a nontrivial finite quadratic variation and the forward
integral of some adapted processes with respect to it coincides with "It\^o's
integral", but it is not a semimartingale. Moreover, some generalized It\^o's
formulas and Bouleau-Yor identities are introduced.
