Weak Uniqueness for the Stochastic Heat Equation Driven by a Multiplicative Stable Noise

We consider the stochastic heat equation $$\frac{\mathrm{\partial }{Y}_t(x)}{\mathrm{\partial }t}=\frac{1}{2}{\mathrm{\Delta }}_x{Y}_t(x)+Y_{t-}(x{)}^{\beta }{\dot{L}}^{\alpha }$$ with $t \ge 0$,$x \in $\mathbb{R}$$and$L^{\alpha}$beingan$\alpha$-stable white noise without negative jumps. Under appropriate non-negative initial conditions, when $\alpha \in (1,2)$ and $\beta \in (\frac{1}{\alpha },1)$ we prove that weak uniqueness holds for the above using the approximating duality approach developed by Mytnik (Ann. Probab. (1998) 26 968-984).