A Local-Time Correspondence for Stochastic Partial Differential Equations

**Abstract.**
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 bigger 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 study mainly 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 that the solutions are unbounded, they
almost surely blow up at exactly the same points.

**Keywords.**
Stochastic heat equation, stochastic wave equation, Gaussian noise,
existence of process solutions, local times, isomorphism theorems

**
AMS Classification (2000).**
Primary. 60H15, 60J55; Secondary. 35R60, 35D05

**Support.** The research of DK was supported in part by a grant from
the U.S. National Science Foundation.

**Pre/E-Prints.** This paper is available in

Mohammud Foondun Department of Mathematics University of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090, U.S.A. mohammud@math.utah.edu |
Davar Khoshnevisan Department of Mathematics University of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090, U.S.A. davar@math.utah.edu |
Eulalia Nualart Institut Galilée Université Paris 13 93430 Villetaneuse, France nualart@math.univ-paris13.fr |

*Last Update: November 26, 2007*

©
2007 - Mohammud Foondun, Davar Khoshnevisan, and Eulalia Nualart