研究期間：10111~10207;There are two common approaches to stochastic partial differential equations.One is infinitely dimensional approach, the other one invented by J. Walsh is more probabilistic. We are using Walsh’s approach and consider stochastic heat equations (SHEs) which is a family of heat equations with added Gaussian noises. Since rough noises have been added, the behaviors of solutions are different from the [deterministic] heat equations (HEs). For example, in dimension one with constant initial data, the solutions to SHEs have fluctuations. Our plan is to understand the behaviors of the solutions, such as intermittency, fractal-like exceedance sets. Moreover, by observation of the solutions in mild form, intuitively the solutions can be locally approximated by the solutions to stochastic differential equations. We are interested in giving a rigorous proof of it. When the initial data vanishes at infinity, the solutions to HEs go to infinity exponentially; in the case of SHEs, we like to know how fast the solutions to SHEs dissipate.