Definition of the Subject
Several nonlinear partial differential evolution equations can be written as abstract equations in a suitable Hilbert space H (we shall denote by | ⋅ | the norm and by ⟨⋅,⋅⟩ the scalar product in H) of the following form (Temam 1988):
where A:D(A) ⊂ H → H is the infinitesimal generator of a strongly continuous semigroup etA of linear-bounded operators in H and b:D(b) ⊂ H → H is a nonlinear mapping.
In order to take into account unpredictable random perturbations, one is led to add to Eq. 1 a term of the form σ(X(t))dW(t), where σ:D(σ) ⊂ H → L(H) (L(H) is the space of all linear operators from H into itself) and W(t) a cylindrical Wiener process in H. Then Eq. 1 is replaced by the following stochastic partial differential equation (SPDEs):
Abbreviations
- Brownian motion:
-
A real Brownian motion B = (B(t)) t≥0 on (Ω, ℱ, ℙ) is a continuous real stochastic process in [0, + ∞) such that (1) B(0) = 0 and for any 0 ≤ s < t, B(t) − B(s) is a real Gaussian random variable with mean 0 and covariance t − s, and (2) if 0 < t 1 < ⋯ < t n , the random variables, B(t 1), B(t 2) − B(t 1), …, B(t n ) − B(t n−1) are independent.
- Continuous stochastic process:
-
Let (Ω, ℱ, ℙ) be a probability space. A continuous stochastic process (with values in H) is a family of (H-valued) random variables (X(t) = X(t, ω)) t≥0 (ω∈Ω) such that X(⋅, ω) is continuous for ℙ – almost all ω∈Ω.
- Cylindrical Wiener process:
-
A cylindrical Wiener process in a Hilbert space H is a process of the form
$$ W(t)={\displaystyle \sum_{k=1}^{\infty }{e}_k{\beta}_k(t),}\kern1em t\ge 0, $$where (e k ) is a complete orthonormal system in H and (β k ) a sequence of mutually independent standard Brownian motions in a probability space (Ω, ℱ, ℙ).
- Evolution equations:
-
Let H be a Hilbert (or Banach) space, T > 0, and f a mapping from [0, T] × H into H. An equation of the form
$$ {u}^{\prime }(t)=f\left(t,u(t)\right),\kern1em t\in \left[0,T\right],\kern2em \left(*\right) $$is called an abstract evolution equation. If H is finite-dimensional, we call ( * ) an ordinary evolution equation, whereas if H is infinite-dimensional and f is a differential operator, we call ( * ) a partial differential evolution equation.
- Noise:
-
Let W(t) be a cylindrical Wiener process in a Hilbert space H. A noise is an expression of the form BW(t), where B∈L(H). If B = I (the identity operator in H), the noise is called white.
- Stochastic dynamical system:
-
This is a system governed by a partial differential evolution equation perturbed by noise.
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Da Prato, G. (2013). Nonlinear Stochastic Partial Differential Equations. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-3-642-27737-5_367-3
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