Logarithmic asymptotics of the densities of SPDEs driven by spatially
correlated noise
release_lapg5osjbbexdliiujr64h3mhe
by
Marta Sanz-Solé, André Süß
2014
Abstract
We consider the family of stochastic partial differential equations indexed
by a parameter ∈(0,1],
Lu^(t,x) =
σ(u^(t,x))Ḟ(t,x)+b(u^(t,x)),
(t,x)∈(0,T]× with suitable initial conditions. In this equation,
L is a second-order partial differential operator with constant coefficients,
σ and b are smooth functions and Ḟ is a Gaussian noise, white
in time and with a stationary correlation in space. Let p^_t,x denote
the density of the law of u^(t,x) at a fixed point
(t,x)∈(0,T]×. We study the existence of _↓ 0^2 p^_t,x(y) for a fixed y∈. The results apply to a class
of stochastic wave equations with d∈{1,2,3} and to a class of stochastic
heat equations with d>1.
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