Jin Feng: Change-of-coordinate methods for SPDEs

We explore some new ideas of defining solution for stochastic PDEs, with a goal of developing well-posedness in the future. 

Among generic methods for understanding deterministic nonlinear PDEs, the approach of variational inequalities is especially important. It is natural and effective whenever the associated differential operator has monotonicity. In such cases, the meaning of an equation informally written as EQUALITY is indeed rigorously understood by a family of INEQUALITIES. The definition of viscosity solution for Hamilton-Jacobi PDEs is such an example. 

When a physical phenomenon (supposedly) described by a PDE or stochastic PDE exhibit non-smooth (time-space) behaviors, change of variable formula for the solution may fail. Hence knowing the solution does not imply knowing functional transforms of the solution. While defining solutions, throwing in additional coordinates to include transforms of other coordinates becomes useful. DiPerna-Lions’ renormalized solution theory is such an example. 

Finally, in describing stochastic nonlinear PDEs, it is helpful to remove a layer of fast scale oscillation before describing the evolution at a slower oscillation scale. We can first define some stochastic linear PDEs, then use them to generate reference stochastic coordinates and define the more difficult nonlinear stochastic PDE.  This is the same principle for the following: if we want to understand trajectories of a collection of bullets shooting in the same direction, it is easier to see fine details by standing on one of the bullets than staring at the bullets from the ground. 

Given an informally written nonlinear stochastic PDE, each of the above three corresponds to a different type of renormalization. In the context of a stochastic Hamilton-Jacobi equation, I illustrate how a notion of solution can be introduced by following these three steps.