Solution To Stochastic Heat Equation Research Articles

Fluctuations of two-dimensional stochastic heat equation and KPZ equation in subcritical regime for general initial conditions

The Kardar-Parisi-Zhang equation (KPZ equation) is solved via Cole-Hopf transformation h=logu, where u is the solution of the multiplicative stochastic heat equation(SHE). In [CD20, CSZ20, G20], they consider the solution of two dimensional KPZ equation via the solution uε of SHE with the flat initial condition and with noise which is mollified in space on scale in ε and its strength is weakened as βε=βˆ 2π −logε, and they prove that when βˆ∈(0,1), 1 βε(loguε−E[loguε]) converges in distribution as a random field to a solution of Edwards-Wilkinson equation. In this paper, we consider a stochastic heat equation uε with a general initial condition u0 and its transformation F(uε) for F in a class of functions F, which contains F(x)=xp (0<p≤1) and F(x)=logx. Then, we prove that 1 βε(F(uε(t,⋅))−E[F(uε(t,⋅))]) converges in distribution as a random field to a centered Gaussian field jointly in finitely many F∈F, t, and u0. In particular, we show the fluctuations of solutions of stochastic heat equations and KPZ equations jointly converge to solutions of SPDEs which depend on u0. Our main tools are Itô’s formula, the martingale central limit theorem, and the homogenization argument as in [CNN22]. To this end, we also prove a local limit theorem for the partition function of intermediate disorder 2d directed polymers.

Weak intermittency of stochastic heat equation under discretizations

Intermittency arises typically in random fields of multiplicative type like the solution of stochastic heat equation. This paper investigates whether the stochastic heat equation with multiplicative noise under a discretization could inherit the dynamical behavior in particular the weak intermittency of the original equation. The exact solution of the stochastic heat equation with multiplicative noise is proved to admit weak intermittency with a specific index of Lyapunov exponents. We prove the existence of the weak intermittency for stochastic heat equation under a class of discretizations, and further the preservation of the index of Lyapunov exponents of the exact solution by a renewal approach, provided in addition that the initial datum is a positive constant and the spatial partition number is large.

Numerical Solution of Stochastic Heat Equation

This paper will use the numerical methods to solve the stochastic heat equation driven by (Brownian motion, Brownian bridge, and Reflected Brownian motion). The numerical schema depends on the impersonation of the equation solution and includes a stochastic portion of the noise and a partial differential equation. We will apply our methods using daily temperatures in Iraq for the year 2018.

Transportation inequalities for stochastic heat equations

Using the method of Girsanov transformation, we establish the transportation inequalities, on the continuous paths space with respect to the L2 metric, for the law of the solution of stochastic heat equation driven by space–time white noise. The case of stochastic heat equations driven by fractional noise is also investigated.

A Note on a Feynman-Kac-Type Formula

In this article, we establish a probabilistic representation for the second-order moment of the solution of stochastic heat equation, with multiplicative noise, which is fractional in time and colored in space. This representation is similar to the one given in Dalang, Mueller and Tribe (2008) in the case of an s.p.d.e. driven by a Gaussian noise, which is white in time. Unlike the formula of Dalang, Mueller and Tribe (2008) ,which is based on the usual Poisson process, our representation is based on the planar Poisson process, due to the fractional component of the noise.

Tail estimates for positive solutions of stochastic heat equation

In this paper, we study the solution of a class of stochastic heat equations of convolution type. We give an explicit solution X t using two basic tools: the characterization theorem for generalized functions and the convolution calculus. For positive initial condition f and coefficients processes Vt, Mt, we prove that the corresponding solution X t admits an integral representation by a certain measure. Finally, we compute the tail estimate for the obtained solution and its expectation.