BMO martingales and positive solutions of heat equations (original) (raw)

On the fundamental solution of heat and stochastic heat equations

arXiv: Analysis of PDEs, 2019

We consider the generic divergence form second order parabolic equation with coefficients that are regular in the spatial variables and just measurable in time. We show that the spatial derivatives of its fundamental solution admit upper bounds that agree with the Aronson type estimate and only depend on the ellipticity constants of the equation and the L ∞ norm of the spatial derivatives of its coefficients. We also study the corresponding stochastic partial differential equations and prove that under natural assumptions on the noise the equation admits a mild solution, given by anticipating stochastic integration.

Regularity and Strict Positivity of Densities for the Nonlinear Stochastic Heat Equation

Memoirs of the American Mathematical Society, 2021

In this paper, we establish a necessary and sufficient condition for the existence and regularity of the density of the solution to a semilinear stochastic (fractional) heat equation with measure-valued initial conditions. Under a mild cone condition for the diffusion coefficient, we establish the smooth joint density at multiple points. The tool we use is Malliavin calculus. The main ingredient is to prove that the solutions to a related stochastic partial differential equation have negative moments of all orders. Because we cannot prove u ( t , x ) ∈ D ∞ u(t,x)\in \mathbb {D}^\infty for measure-valued initial data, we need a localized version of Malliavin calculus. Furthermore, we prove that the (joint) density is strictly positive in the interior of the support of the law, where we allow both measure-valued initial data and unbounded diffusion coefficient. The criteria introduced by Bally and Pardoux are no longer applicable for the parabolic Anderson model. We have extended thei...