| Abstract: |
| Consider $[0,1]$-valued random field solution $(u_t(x))_{t\geq 0, x\in \mathbb R}$ to the one-dimensional stochastic heat equation \[ \partial_t u_t = \frac{1}{2}\Delta u_t + b(u_t) + \sqrt{u_t(1-u_t)} \dot W \] where $b(1)\leq 0\leq b(0)$ and $\dot W$ is a space-time white noise. In this talk, we present the weak existence and uniqueness of the above equation for a class of drifts $b(u)$ that may be irregular at the points where the noise is degenerate, that is, at $u=0$ or $u=1$. This class of drifts includes non-Lipschitz drifts like $b(u) = u^q(1-u)$ for every $q\in (0,1)$, and some discontinuous drifts like $b(u) = \mathbf 1_{(0,1]}(u)-u$. This demonstrates a regularization effect of the multiplicative space-time white noise without assuming the standard assumption that the noise coefficient is Lipschitz and non-degenerate.
The method we apply is a further development of a moment duality technique that uses branching-coalescing Brownian motions as the dual particle system. To handle an irregular drift in the above equation, particles in the dual system are allowed to have a number of offspring with infinite expectation, even an infinite number of offspring with positive probability. We show that, even though the branching mechanism with infinite number of offspring causes explosions in finite time, immediately after each explosion the total population comes down from infinity due to the coalescing mechanism. Our results on this dual particle system are of independent interest. |
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