Global solution of nonlinear stochastic heat equation with solutions in a Hilbert manifold

2020 ◽  
Vol 20 (06) ◽  
pp. 2040012
Author(s):  
Zdzisław Brzeźniak ◽  
Javed Hussain
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Hilbert Space ◽  
Evolution Equation ◽  
Heat Equation ◽  
Global Solution ◽  
Unit Sphere ◽  
Stochastic Partial Differential Equation ◽  
Hilbert Manifold ◽  
Unconstrained Problem

The objective of this paper is to prove the existence of a global solution to a certain stochastic partial differential equation subject to the [Formula: see text]-norm being constrained. The corresponding evolution equation can be seen as the projection of the unconstrained problem onto the tangent space of the unit sphere [Formula: see text] in a Hilbert space [Formula: see text].

scholarly journals Controllability of semilinear stochastic evolution equations in Hilbert space

2001 ◽  
Vol 14 (4) ◽  
pp. 329-339 ◽  
Author(s):  
P. Balasubramaniam ◽  
J. P. Dauer
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Evolution Equations ◽  
Stochastic Partial Differential Equation ◽  
Semigroup Theory ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations

Controllability of semilinear stochastic evolution equations is studied by using stochastic versions of the well-known fixed point theorem and semigroup theory. An application to a stochastic partial differential equation is given.

A non-conservation stochastic partial differential equation driven by anisotropic fractional Lévy random field

2020 ◽  
pp. 2150028
Author(s):  
Xuebin Lü ◽  
Wanyang Dai
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Equation ◽  
Random Field ◽  
Stochastic Partial Differential Equation ◽  
Partial Differential ◽  
Linear Heat Equation ◽  
Linear Heat ◽  
Sample Paths ◽  
Special Cases

We study a non-conservation second-order stochastic partial differential equation (SPDE) driven by multi-parameter anisotropic fractional Lévy noise (AFLN) and under different initial and/or boundary conditions. It includes the time-dependent linear heat equation and quasi-linear heat equation under the fractional noise as special cases. Unique existence and expressions of solution to the equation are proved and constructed. An AFLN is defined as the derivative of an anisotropic fractional Lévy random field (AFLRF) in certain sense. Comparing with existing noise systems, our non-Gaussian fractional noises are essentially observed from random disturbances on system accelerations rather than from those on system moving velocities. In the process of proving our claims, there are three folds. First, we consider the AFLRF as the generalized functional of sample paths of a pure jump Lévy process. Second, we build Skorohod integration with respect to the AFLN by white noise approach. Third, by combining this noise analysis method with the conventional PDE solution techniques, we provide solid proofs for our claims.

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