Heat equation with strongly inhomogeneous noise

The inhomogeneous heat equation on T

Define ‖f‖Hk =  ∑ 0≤j≤k ∥∥∂j xf∥∥2L2 1/2 . If u is a distribution on T, ∂xu is also a distribution on T, and in particular, if u ∈ L(T) then ∂xu is a distribution on T. But if u ∈ H(T), for example, then ∂ xu is an element of L (T), rather than merely being a distribution. Fix T > 0. Let f ∈ L(0, T ;L(T)) and g ∈ H(T); as H(T) ⊂ C(T), we can speak about the value of g at every point rather ...

Stochastic heat equation with white-noise drift

A Quasilinear Parabolic Equation With Inhomogeneous Density and Absorption

Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

We consider the stochastic heat equation with multiplicative noise ut = 1 2 ∆u + uẆ in R+ × R , whose solution is interpreted in the mild sense. The noise Ẇ is fractional in time (with Hurst index H ≥ 1/2), and colored in space (with spatial covariance kernel f). When H > 1/2, the equation generalizes the Itô-sense equation for H = 1/2. We prove that if f is the Riesz kernel of order α, or the ...

Stochastic Heat Equation with Infinite Dimensional Fractional Noise: L2-theory

In this article we consider the stochastic heat equation in [0, T ]× Rd, driven by a sequence (β)k of i.i.d. fractional Brownian motions of index H > 1/2 and random multiplication functions (g)k. The stochastic integrals are of Hitsuda-Skorohod type and the solution is interpreted in the weak sense. Using Malliavin calculus techniques, we prove the existence and uniqueness of the solution in a ...