Random Heat Equation Research Articles

The Random Heat Equation in Dimensions Three and Higher: The Homogenization Viewpoint

We consider the stochastic heat equation $$\partial _{s}u =\frac{1}{2}\Delta u +(\beta V(s,y)-\lambda )u$$ ∂ s u = 1 2 Δ u + ( β V ( s , y ) - λ ) u , with a smooth space-time-stationary Gaussian random field V(s, y), in dimensions $$d\ge 3$$ d ≥ 3 , with an initial condition $$u(0,x)=u_0(\varepsilon x)$$ u ( 0 , x ) = u 0 ( ε x ) and a suitably chosen $$\lambda \in {\mathbb {R}}$$ λ ∈ R . It is known that, for $$\beta $$ β small enough, the diffusively rescaled solution $$u^{\varepsilon }(t,x)=u(\varepsilon ^{-2}t,\varepsilon ^{-1}x)$$ u ε ( t , x ) = u ( ε - 2 t , ε - 1 x ) converges weakly to a scalar multiple of the solution $${\bar{u}}(t,x)$$ u ¯ ( t , x ) of the heat equation with an effective diffusivity a, and that fluctuations converge, also in a weak sense, to the solution of the Edwards-Wilkinson equation with an effective noise strength $$\nu $$ ν and the same effective diffusivity. In this paper, we derive a pointwise approximation $$w^\varepsilon (t,x)={\bar{u}}(t,x)\Psi ^\varepsilon (t,x)+ \varepsilon u_1^\varepsilon (t,x)$$ w ε ( t , x ) = u ¯ ( t , x ) Ψ ε ( t , x ) + ε u 1 ε ( t , x ) , where $$\Psi ^\varepsilon (t,x)=\Psi (t/\varepsilon ^2,x/\varepsilon )$$ Ψ ε ( t , x ) = Ψ ( t / ε 2 , x / ε ) , $$\Psi $$ Ψ is a solution of the SHE with constant initial conditions, and $$u^\varepsilon _1$$ u 1 ε is an explicit corrector. We show that $$\Psi (t,x)$$ Ψ ( t , x ) converges to a stationary process $${\tilde{\Psi }}(t,x)$$ Ψ ~ ( t , x ) as $$t\rightarrow \infty $$ t → ∞ , that $${\mathbf {E}}|u^\varepsilon (t,x)-w^\varepsilon (t,x)|^2$$ E | u ε ( t , x ) - w ε ( t , x ) | 2 converges pointwise to 0 as $$\varepsilon \rightarrow 0$$ ε → 0 , and that $$\varepsilon ^{-d/2+1}(u^\varepsilon -w^\varepsilon )$$ ε - d / 2 + 1 ( u ε - w ε ) converges weakly to 0 for fixed t. As a consequence, we derive new representations of the diffusivity a and effective noise strength $$\nu $$ ν . Our approach uses a Markov chain in the space of trajectories introduced in [17], as well as tools from homogenization theory. The corrector $$u_1^\varepsilon (t,x)$$ u 1 ε ( t , x ) is constructed using a seemingly new approximation scheme on a mesoscopic time scale.

Well-posedness for the coupling of a random heat equation with a multiplicative stochastic Barenblatt equation

In this contribution, a stochastic nonlinear evolution system under Neumann boundary conditions is investigated. Precisely, we are interested in finding an existence and uniqueness result for a random heat equation coupled with a Barenblatt’s type equation with a multiplicative stochastic force in the sense of Itô. In a first step, we establish well-posedness in the case of an additive noise through a semi-implicit time discretization of the system. In a second step, the derivation of continuous dependence estimates of the solution with respect to the data allows us to show the desired existence and uniqueness result for the multiplicative case.

Investigation of sample paths properties for some classes of φ-sub-Gaussian stochastic processes

This paper investigates sample paths properties of φ-sub-Gaussian processes by means of entropy methods. Basing on a particular entropy integral, we treat the questions on continuity and the rate of growth of sample paths. The obtained results are then used to investigate the sample paths properties for a particular class of φ-sub-Gaussian processes related to the random heat equation. We derive the estimates for the distribution of suprema of such processes and evaluate their rate of growth.

On average controllability of random heat equations with arbitrarily distributed diffusivity

In this paper, we study the average controllability of a random heat equation, with the diffusivity serving as the random variable drawn from a general probability distribution. We show that the solutions of such random heat equations are both null and approximately controllable in average from an arbitrary open set of the domain and in an arbitrarily small time, recovering the known result when the random diffusivity is uniformly or exponentially distributed.

Uncertainty quantification for random parabolic equations with nonhomogeneous boundary conditions on a bounded domain via the approximation of the probability density function

This paper deals with the randomized heat equation defined on a general bounded interval [L1, L2] and with nonhomogeneous boundary conditions. The solution is a stochastic process that can be related, via changes of variable, with the solution stochastic process of the random heat equation defined on [0,1] with homogeneous boundary conditions. Results in the extant literature establish conditions under which the probability density function of the solution process to the random heat equation on [0,1] with homogeneous boundary conditions can be approximated. Via the changes of variable and the Random Variable Transformation technique, we set mild conditions under which the probability density function of the solution process to the random heat equation on a general bounded interval [L1, L2] and with nonhomogeneous boundary conditions can be approximated uniformly or pointwise. Furthermore, we provide sufficient conditions in order that the expectation and the variance of the solution stochastic process can be computed from the proposed approximations of the probability density function. Numerical examples are performed in the case that the initial condition process has a certain Karhunen‐Loève expansion, being Gaussian and non‐Gaussian.

The Edwards–Wilkinson Limit of the Random Heat Equation in Dimensions Three and Higher

We consider the heat equation with a multiplicative Gaussian potential in dimensions $d\geq 3$. We show that the renormalized solution converges to the solution of a deterministic diffusion equation with an effective diffusivity. We also prove that the renormalized large scale random fluctuations are described by the Edwards-Wilkinson model, that is, the stochastic heat equation (SHE) with additive white noise, with an effective variance.

Random Crank-Nicolson Scheme for Random Heat Equation in Mean Square Sense

The goal of computational science is to develop models that predict phenomena observed in nature. However, these models are often based on parameters that are uncertain. In recent decades, main numerical methods for solving SPDEs have been used such as, finite difference and finite element schemes [1]-[5]. Also, some practical techniques like the method of lines for boundary value problems have been applied to the linear stochastic partial differential equations, and the outcomes of these approaches have been experimented numerically [7]. In [8]-[10], the author discussed mean square convergent finite difference method for solving some random partial differential equations. Random numerical techniques for both ordinary and partial random differential equations are treated in [4] [10]. As regards applications using explicit analytic solutions or numerical methods, a few results may be found in [5] [6] [11]. This article focuses on solving random heat equation by using Crank-Nicol- son technique under mean square sense and it is organized as follows. In Section 2, the mean square calculus preliminaries that will be required throughout the paper are presented. In Section 3, the Crank-Nicolson scheme for solving the random heat equation is presented. In Section 4, some case studies are showed. Short conclusions are cleared in the end section.

Stochastic Burgers PDEs with random coefficients and a generalization of the Cole–Hopf transformation

This paper studies forward and backward versions of the random Burgers equation (RBE) with stochastic coefficients. First, the celebrated Cole–Hopf transformation reduces the forward RBE to a forward random heat equation (RHE) that can be treated pathwise. Next we provide a connection between the backward Burgers equation and a system of FBSDEs. Exploiting this connection, we derive a generalization of the Cole–Hopf transformation which links the backward RBE with the backward RHE and investigate the range of its applicability. Stochastic Feynman–Kac representations for the solutions are provided. Explicit solutions are constructed and applications in stochastic control and mathematical finance are discussed.

Gradient of the log-likelihood ratio for infinite dimensional stochastic systems

Using a covariance operator approach, we derive an expression for the log-likelihood ratio gradient for system parameter estimation for continuous-time infinite-dimensional stochastic systems. The gradient formula includes the smoother estimates and derivatives of system operators, with no derivatives of estimates or covariance operators. The unbounded operators typically found in partial differential equations limit how much the gradient formula can be simplified. A random heat equation is considered.

The origin of scaling in the galaxy distribution

I present an analytic model for non-linear clustering of the luminous (baryonic) material in a universe in which the gravitational field is dominated by dark matter. The model is based on a two-component generalization of the adhesion approximation in which the gravitational potential of the dark component is determined by the standard Zel’dovich approximation or one of its variants, or by an N-body simulation. The baryonic matter flow is dissipative and is driven by this dark matter gravitational potential. The velocity potential of the matter is described by a generalization of the Burgers equation: the random heat equation (‘RH equation’) with a spatially correlated Gaussian driving potential. The properties of the RH equation are well understood: it is closely related to the equation for the Anderson model and to Brownian motion in a random potential: the solution can be expressed in terms of path integrals. Using this it is possible to derive the scaling properties of the solution and, in particular, those of the resultant velocity field. Even though the flow is non-linear, the velocity field remains Gaussian and inherits its scaling properties from the gravitational potential. This provides an underlying dynamical reason why the density field in the baryonic component is lognormally distributed and manifests multifractal scaling. By explicitly putting dark and luminous matter on different footings, the model provides an improved framework for considering the growth of large-scale cosmic structure. It provides a solution for the velocity potential of the baryonic component in closed form (albeit a path integral) from which the statistical properties of the baryonic flow can be derived.

Random heat equation: Solutions by the stochastic adaptive interpolation method

The one-dimensional nonlinear heat equation is considered with stochastic coefficients and initial/boundary values in a finite strip and in the half-space. Approximated solutions are obtained by the stochastic adaptive interpolation method which corresponds first to transforming the original partial differential equation into a system of ordinary differential equations via a generalization to the stochastic case of the Bellman's differential quadrature method. The system of ordinary differential equations is then solved by a continuous approximation following by Adomian's decomposition method, and the solutions are compared with those obtained by more standard numerical techniques.

Solutions to the random heat equation by the method of successive approximations

An iterative scheme for solving the random heat equation is proposed. Convergence of the method is established. Properties of the solution as well as error estimates are obtained. Indications as to possible application to nonlinear, inhomogeneous, time-dependent, random diffusion problems are given. A specific example of application to random diffusion in the unit interval is treated both analytically and numerically.