On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

By Daniel Conus, Davar Khoshnevisan. 2010

67 citations recorded by the source corpus.

We study the stochastic heat equation $${\partial_t u = \mathcal{L}u+\sigma(u)\dot W}$$ in (1 + 1) dimensions, where $${\dot W}$$ is space-time white noise, σ : R → R is Lipschitz continuous, and $${\mathcal{L}}$$ is the generator of a symmetric Lévy process that has finite exponential moments, and u0 has exponential decay at ±∞. We prove that under natural conditions on σ : (i) The νth absolute moment of the solution to our stochastic heat equation grows exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. Very little else seems to be known about the location of the high peaks of the solution to the stochastic heat equation under the present setting (see, however, Gärtner et al. in Probab Theory Relat Fields 111:17–55, 1998; Gärtner et al. in Ann Probab 35:439–499, 2007 for the analysis of the location of the peaks in a different model). Finally, we show that these results extend to the stochastic wave equation driven by Laplacian.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d7166c5ab33426534bbdea4f5b84c4503a673434

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

In Mathematical Physics

Citations in the corpus

Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions (2013, background): “When the initial data is constant, these two exponents do not depend on x.”
Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency (2014, background): “g [12, 13, 14, 18, 31]), and as a localization property for the polymer measure [7].”