Weak Moment of a Class of Stochastic Heat Equation with Martingale-valued Harmonic Function

A study of a non-linear parabolic SPDEs of the form YY2.PNGwith XX.PNG as the space-time white noise and XXXX2.PNGa space-time harmonic function was done. The function XXX3.PNG is Lipschitz continuous and XXXX3.PNG the XXX4.PNG -generator of a Lévy process . Some precise condition for existence and uniqueness of the solution were given and we show that the solution grows weakly(in law/distribution) in time (for large ) at most a precise exponential rate for the XXX7.PNG ; and grows in time at most a precise exponential rate for the case of XXX5.PNG generator of an alpha-stable process .