A local-time correspondence for stochastic partial differential equations

It is frequently the case that a white-noise-driven parabolic and/or hyperbolic stochastic partial differential equation (SPDE) can have random-field solutions only in spatial dimension one. Here we show that in many cases, where the ``spatial operator'' is the $L^2$-generator of a Lévy process $X$, a linear SPDE has a random-field solution if and only if the symmetrization of $X$ possesses local times. This result gives a probabilistic reason for the lack of existence of random-field solutions in dimensions strictly larger than one.

In addition, we prove that the solution to the SPDE is [Hölder] continuous in its spatial variable if and only if the said local time is [Hölder] continuous in its spatial variable. We also produce examples where the random-field solution exists, but is almost surely unbounded in every open subset of space-time. Our results are based on first establishing a quasi-isometry between the linear $L^2$-space of the weak solutions of a family of linear SPDEs, on one hand, and the Dirichlet space generated by the symmetrization of $X$, on the other hand.

We mainly study linear equations in order to present the local-time correspondence at a modest technical level. However, some of our work has consequences for nonlinear SPDEs as well. We demonstrate this assertion by studying a family of parabolic SPDEs that have additive nonlinearities. For those equations we prove that if the linearized problem has a random-field solution, then so does the nonlinear SPDE. Moreover, the solution to the linearized equation is [Hölder] continuous if and only if the solution to the nonlinear equation is, and the solutions are bounded and unbounded together as well. Finally, we prove that in the cases where the solutions are unbounded, they almost surely blow up at exactly the same points.