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Stochastic partial differential equation (changes)

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Contents

Idea

Stochastic partial differential equations are partial differential equations with random processes, much like stochastic ordinary differential equations are ordinary differential equations with random processes. The topic is an active research area, and there exist several different approaches.

An example of a SPDE is the Burgers' equation with stochastic forcing, which is a model for e.g. turbulence.

Details

Nonlinear Evolution Equations, the “Variational Approach”

Solutions of SPDE are stochastic processes in continuous time in infinite dimensional spaces like spaces of (generalized) functions, Banach and Hilbert spaces or infinite dimensional manifolds, depending on the approach taken.

On this page we will concentrate on the stochastic Burgers' equation and the stochastic Navier-Stokes equations. Both are examples of nonlinear evolution equations, with a notion of a time variable and spatial variables. Both equations can be handled with Hilbert spaces as solution spaces. We fix two separable, real Hilbert spaces U,HU, H. Every integrals in Banach spaces on this page are Bochner integrals.

The Itô Integral: Construction Needed

In a first step we will need to construct the stochastic Itô integral

0 tϕ(s)dW s,t[0,T] \int_0^t \phi(s) d W_s, \; t \in [0, T]

where WW is a Wiener process on UU, and ϕ\phi is a process which takes values in L(U,H)L(U, H), the space of unbounded linear operators from UU to HH.

For a topological space XX let 𝔹(X)\mathbb{B}(X) be its σ\sigma-algebra.

Gaussian Random Variable

Random variables are functions on a probability space (Ω,F,P)(\Omega, F, P), (set of events, σ\sigma-algebra, measure), which we will mostly suppress as is commonly done in the literature.

Definition

A probability measure μ\mu on (U,𝔹(U))(U, \mathbb{B}(U)) is called a Gaussian probability measure if for all uUu \in U we have that the mapping

u:U,vu,v u: U \to \mathbb{R}, \; v \mapsto \langle u, v \rangle

is a Gaussian real random variable.

As in the finite dimensional situation, Gaussian probability measures are completely characterized by their characteristic function(al), this is the Bochner-Minlos theorem.

Proposition

A measure μ\mu on (U,𝔹(U))(U, \mathbb{B}(U)) is a Gaussian probability measure iff

μ^(u):= Ue iu,vμ(dv)=e im,u12Qu,u \hat \mu (u) := \int_U e^{i \langle u, v \rangle} \mu (d v) = e^{i \langle m, u \rangle - \frac{1}{2} \langle Q u, u \rangle}

for all uUu \in U, with a mUm \in U (the mean) and QQ a selfadjoint, positive operator of trace class (the covariance operator).

We will say that a UU-valued Gaussian random variable XX is a N(m,Q)N(m, Q) random variable just like in the finite dimensional case.

Since QQ is selfadjoint and of trace class, it has a orthonormal basis of eigenvectors e k,ke_k, k \in \mathbb{N}, this is the Hilbert-Schmidt theorem:

Qe k=λ ke k Q \; e_k = \lambda_k \; e_k

with 00 as the only accumulation point of the sequence (λ k)(\lambda_k).

It is possible to characterize N(m,Q)N(m, Q) random variables as follows:

Proposition

A UU-valued random variable is a N(m,Q)N(m, Q) random variable iff

X= kλ kβ ke k+m X = \sum_{k \in \mathbb{N}} \sqrt{\lambda_k} \; \beta_k \; e_k + m

with independent, real N(0,1)N(0, 1) random variables β k\beta_k, the sum converges in the L 2L^2 sense.

Let’s note as a remark that we can construct in this way a N(m,Q)N(m, Q) random variable for every mm and every selfadjoint, positive, trace-class operator QQ. We are now ready to define the standard QQ-Wiener process given such an operator QQ:

Wiener Process

As usual we will mostly suppress the underlying probability space (Ω,F,P)(\Omega, F, P), (set of events, σ\sigma-algebra, measure).

Definition

A UU-valued random process W tW_t is called a standard QQ-Wiener process iff

  • W(0)=0W(0) = 0,

  • WW has a.s. continuous trajectories,

  • the increments are independent and

  • the increments W tW sW_t - W_s are N(0,(ts)Q)N(0, (t-s) Q) random variables with t>st \gt s.

So, the definition of Wiener processes in infinite dimensions is essentially the same as in finite dimensions.

The Q-Wiener process can be characterized similar to Gaussian random variables:

Proposition

A UU-valued random process W tW_t is a QQ-Wiener process iff

W t= kλ kβ k(t)e k W_t = \sum_{k \in \mathbb{N}} \sqrt{\lambda_k} \; \beta_k(t) \; e_k

where the β k(t)\beta_k(t) are real valued Wiener processes. The sum converges in L 2(C([0,T],U),P)L^2(C([0, T], U), P) and has a P-a.s. continuous modification, where the space C([0,T],U)C([0, T], U) has the sup-norm.

As a corollary we see that there is to every suitable covariance operator QQ a QQ-Wiener process.

Martingales in Banach Spaces

We need the concept of martingales in Banach spaces analog to the situation in finite dimensions. In a first step we note that the conditional expectation exists also in Banach spaces:

Proposition

Let E be a real separable Banach space and X be a Bochner-integrable E-valued random variable on the probability space (Ω,F,P)(\Omega, F, P), let GG be a σ\sigma-field contained in FF.

Then there exists a GG-measurable random variable YY such that for all gGg \in G we have:

gXdP= gYdP \int_g X d P = \int_g Y d P

All such random variables are equal P-almost everywhere. As in the finite dimensional case we write Y=E(X|G)Y = E(X | G) and say that YY is a conditional expectation of XX given GG. Further we have:

E(X|G)E(X|G) \| E(X | G) \| \leq E( \| X \| | G)

The definition of a martingale with respect to a filtration is the same as in the finite dimensional case. We can get real-valued submartingales from EE-valued martingales:

Proposition

If M tM_t is an EE-valued martingale and p[1,)p \in [1, \infty), then M t p \| M_t \|^p is a real-valued submartingale.

This means that we can translate Doob’s maximal inequality directly to EE-valued martingales.

Let T 2(E)\mathcal{M}^2_T (E) for a fixed TT be the space of all EE-valued square integrable martingales, then we have the

Proposition

T 2(E)\mathcal{M}^2_T (E) is a Banach space when equipped with the norm

M t T 2:=sup t[0,T](E(M t 2)) 12=(E(M T 2)) 12 \| M_t \|_{\mathcal{M}^2_T} := sup_{t \in [0, T]} (E (\| M_t \|^2) )^{\frac{1}{2}} = (E (\| M_T \|^2) )^{\frac{1}{2}}

Note: Every QQ-Wiener process with respect to a normal filtration (which is defined in the same way as in the finite dimensional case) is a continuous square integrable martingale.

Itô Integral: The Definition

The definition of the Itô integral proceeds essentially in the same way as in the finite dimensional case, but one has to do some extra exercises in linear functional analysis along the way.

…to be continued.

References

Most content of this page is based on:

  • Claudia Prévôt, Michael Röckner: A concise course on stochastic partial differential equations. (ZMATH)

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