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

We consider the stochastic heat equation $$\frac{\partial Y_t(x)}{\partial t} = \frac{1}{2} \Delta_x Y_t(x) + Y_{t-}(x)^{\beta} \dot{L}^{\alpha}$$ with $t \ge 0$, $x \in \mathbb{R}$ and $L^{\alpha}$ being an $\alpha$-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).