Stochastic heat equation with rough dependence in space

Non-linear Rough Heat Equation

This article is devoted to define and solve an evolution equation of the form dyt = ∆yt dt + dXt(yt), where ∆ stands for the Laplace operator on a space of the form L(R), and X is a finite dimensional noisy nonlinearity whose typical form is given by Xt(φ) = ∑N i=1 x i tfi(φ), where each x = (x , . . . , x) is a γ-Hölder function generating a rough path and each fi is a smooth enough function d...

Stochastic heat equation with white-noise drift

The Fourier Heat Equation in Riemannian Space.

1. Let a certain body, capable of absorbing heat, occupy a region of n dimensional Riemannian space V, whose metric is defined by the positive quadratic differential form ds2 = gjj dxl dxJ. Suppose that this body is heated by conduction in any manner. The situation can be visualized as a flow of heat from the warmer to the cooler parts of the body. The rate of flow of heat within the body can b...

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 ...

Moments and Growth Indices for the Nonlinear Stochastic Heat Equation with Rough Initial Conditions1 by Le Chen

We study the nonlinear stochastic heat equation in the spatial domain R, driven by space–time white noise. A central special case is the parabolic Anderson model. The initial condition is taken to be a measure on R, such as the Dirac delta function, but this measure may also have noncompact support and even be nontempered (e.g., with exponentially growing tails). Existence and uniqueness of a r...