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

TL;DR

This paper proves weak uniqueness for a stochastic heat equation driven by multiplicative stable noise, using an approximating duality approach, under specific conditions on the noise stability index and the nonlinearity exponent.

Contribution

It introduces a novel application of the duality approach to establish weak uniqueness for the stochastic heat equation with stable noise.

Findings

Weak uniqueness holds for the equation under given conditions.

The duality approach effectively handles stable noise without negative jumps.

Results extend understanding of stochastic PDEs driven by stable noise.

Abstract

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).