Brownian Motion

Notes from Schilling, Partzsch.

Robert Brown’s new thing

A particle

• starts at \(x=0\).
• at times \(k\De t\), move \(\De x\) units to L or R with equal probability.

Let \(X_t\) be position at time \(t\in [0,T]\).

• \(\Var(X_t)=\si^2 T\) where \(\fc{(\De x)^2}{\De t}=\si^2\).
• CLT: \(X_T \to \sqrt T \si N(0,1)\).

Definition. A \(d\)-dimensional Brownian motion \(B=(B_t)_{t\ge 0}\) is a stochastic process indexed by \([0,\iy)\), \[\begin{align} B_0(\om)&=0\\ B_{t_n}-B_{t_{n-1}},\ldots, B_{t_1}-B_{t_0} & \text{ independent}, 0=t_0\le t_1<\cdots < t_n\\ B_t - B_s &\sim B_{t+h} - B_{s+h}\\ B_t &= N(0,t)^{\ot d}\\ t&\mapsto B_t(\om) \text{ continuous.} \end{align}\]

Note the last property is implied by the previous.

Brownian motion as a Gaussian process

Characteristic function of Gaussian: \[ \E e^{i\an{\xi,\Ga}} = e^{i \E\an{\xi, \Ga} - \rc \Var\an{\xi, \Ga}} = e^{i\an{\xi, \Ga}} = e^{i\an{\xi, m} - \rc 2\an{\xi, \Si\xi}}. \]

• \(B_t\) is Gaussian with mean 0, variance \(t\). \(\E e^{\ze B_t} = e^{t\ze^2/2}\).
• \(\E (B_t^{2k}) = t^k \fc{2^k \Ga(k+\rc 2)}{\sqrt \pi}\).
• \(\Cov(B_s,B_t) = \min(s,t)\).

\((X_t)_{t\ge 0}\) is a Gaussian process if for all \(0\le t_1<t_2<\cdots\), \((X_{t_1},\ldots, X_{t_n})\) is a Gaussian random vector.

Theorem. \((B_t)_{t\ge 0}\) is a Gaussian process. The covariance of \((B_{t_1},\ldots, B_{t_n})\) is \(C = (\min(t_j,t_k))_{j,k}\).

Proof. Linearly transform \((t_1-t_0,t_2-t_1,\ldots)\).

Converse: If the covariance matrices are given by the above and \((X_t)_{t\ge 0}\) has continuous sample paths, then \((X_t)_{t\ge 0}\) is 1-D Brownian.

2.2 Invariance properties

• Reflection \(-B_t\)
• Renewal \(W(t) = B(t+a)-B(a)\).
• Markov property: \(B(t):0\le t\le a\) independent to \(W(t), t\ge 0\).
• Time inversion \(W_t=B_{a-t}-B_a\).
• Scaling \(B_{ct}\sim c^{\rc 2}B_t\).
• Projective reflection \(W(t) = tB(\rc t)\)

2.3 \(\R^d\)

\(B_t\) is \(BM^d\) iff its coordinates are \(BM^1\).

Proof. Forward: Show increments are independent—characteristic function factorizes.

\(Q\)-Brownian motion: \(X_t-X_s\sim N(0,(t-s)Q)\), \(s<t\).

3 Constructions of Brownian motion

3.1 Levy-Ciesielski

Write paths as random series wrt complete orthonormal system of \(L^2([0,1],dt)\). Let \((G_n)_{n\ge 0}\) be sequence of \(N(0,1)\) variables. \[ W_N(t):=\sumz n{N-1} G_n\an{\one_{[0,t)}, \phi_n}_{L^2}. \]

Theorem. \(\lim_{N\to \iy} W_N(t)\) is Brownian motion.

Proof.

• Convergence: \(\E [W_N(t)^2] \to t\).
• Covariances correct: \(\E[(W(t)-W(s))(W(v)-W(u))] = \an{\one_{[s,t)}, \one_{[u,v)}}\).
• Continuity of \(t\mapsto W(t,\om)\): Proof for a specific system: Haar functions \(H_n\).
  • (? Uniform convergence \(\liminf_{n,N\to \iy} \ve{W_{2^N}-W_{2^n}}_{L^{\iy}(t)}=0\). By Arzela-Ascoli (?) there is convergent subsequence.

3.2 Levy’s original argument

4 The canonical model

Identify \(\Om\) (in \((\Om, A, \Pj)\)) as a subset of \((\R^d)^I\), actually \(C_{(0)}\) consisting of continuous \(w:[0,\iy)\to \R^d\), \(w(0)=0\).

Consider the product \(\si\)-algebra \[ B^I(\R^d) = \si\set{\pi_t^{-1}(B)}{B\in B(\R^d), t\in I} = \si\set{\pi_t}{t\in I}. \] Consider the intersection with \(C_{(0)}\), \[ C_{(0)}\cap B^T(\R^d) = \si(\pi_t|_{C_{(0)}}:t\in I). \] \(C_{(0)}\) is complete separable metric space with metric of locally uniform convergence \(\rh(w,v)=\sumo n{\iy} (1\wedge \sup_{0\le t\le n} |w(t)-v(t)|)2^{-n}\).

The finite-dimensional distributions uniquely determine \(\mu\) (cylinder sets generate). \(\mu\) is Wiener measure, the space is the path space.

Theorem \((C_{(0)}, B(C_{(0)}), \mu)\): \((\pi_t)_{t\ge 0}\) is Brownian motion (canonical model of Brownian motion).

Some properties

• \(\Ga\in B^I(\R^d)\) is determined by countably many indices: for every \(\Ga\in B^I(\R^d)\) there exists countable \(S\sub I\), such that \(f\in (\R^d)^I, w\in \Ga, f|_S=w|_S \implies f\in \Ga\). Proof. \(\set{\Ga \subeq (\R^d)^I}{\pat{holds for some countable }S}\) is a \(\si\)-algebra.
• \(C_{(0)}\nin B^{[0,\iy)}\nin B^I (\R^d)\). Proof: can’t enforce continuity using values at countably many points. (What is the point of this? Saying we can’t work in \(B^I(\R^d)\), have to work in \(\cap C_{(0)}\)?)

4.2 Kolmogorov’s Theorem

Theorem. Let \(I\subeq [0,\iy)\), \(p_{t_1,\ldots, t_n}\) be probability measures defined on \((\R^d)^n\). If the family is consistent (\(p_t(C) = p_{t_{\si}} (C_\si)\) and \(p_{t_{1:n-1},t_n}(C_{1:n-1}\times \R^d) = p_{t_{1:n-1}} (C_{1:n-1})\)), then there exists \(\mu\) on \(((\R^d)^I, B^I(\R^d))\), \(p_{t_{1:n}}(C) = \mu(\pi_{t_{1:n}}^{-1}(C))\).

Corollary: can construct canonical process for any family of consistent finite dimensional probability distributions.

(Still not too clear on it sufficing to consider finite-dim projections…)

Can use this theorem to construct BM. Continuity follows from Theorem 4.11.

6 Brownian motion as a Markov process

(I’m confused about what more 6.1 says. Is this related to stopping times? \(F_t\) is info up to time \(t\)?)

Let \[ \Pj^x(B_{t_i}\in A_i:1\le i\le n) = \Pj(B_{t_i}+x\in A_i:1\le i\le n). \]

7 Brownian motion and transition semigroups

Linear operators: transition semigroup and resolvent \[\begin{align} P_t u(x) &= \E^x u(B_t)\\ U_\al u(x) &= \E^x \ba{\iiy u(B_t)e^{-\al t}\,dt} \end{align}\]

A semigroup \((P_t)_{t\ge 0}\) on a Banach space is family of linear operators \(P_t:B\to B, t\ge 0\), satisfying \(P_tP_s=P_{t+s}\), \(P_0=\id\).

Banach spaces:

• \(B_b\): Borel measurable \(\R^d\to \R\) with uniform norm \(\ved_{\iy}\).
• \(C_\iy\): continuous functions vanishing at infinity with uniform norm \(\ved_{\iy}\).

Lemma 7.2. \((B_t)_{t\ge 0}\) is uniformly stochastically continuous, \[ \lim_{t\to 0}\sup_{x\in \R^d} \Pj^x (|B_t-x|>\de)=0\] for all \(\de>0\).

Properties

1. Conservative \(P_t1=1\).
2. Contraction on \(B_b\): \(\ve{P_t u}_\iy \le \ve{u}_{\iy}\), \(u\in B_b\)
3. Positive preserving: \(u\ge 0\implies P_tu\ge 0\), \(u\in B_b\).
4. Sub-Markovian: \(0\le u\le 1\), \(u\in B_b\implies 0\le P_tu\le 1\).
5. Feller: \(u\in C_\iy\implies P_t u\in C_\iy\).
6. Strongly continuous on \(C_\iy\): \(u\in C_\iy\implies \lim_{t\to 0}\ve{P_tu-u}_\iy = 0\).
7. Strong Feller: \(u\in B_b\implies P_tu\in C_b\).

(1-4 is Markov, 2-4 is sub-Markov, 2-6 is Feller, 2-4+7 is strong Feller.)

7.5: \[\Pj^x(B_{t_i}\in C_i:1\le i\le n) = P_{t_1}[\one_{C_1} P_{t_2-t_1}[\one_{C_2}\cdots ]].\]

This gives a way to construct a Markov process from a semigroup of operators. (Apply to indicator functions.) A Feller semigroup has a corresponding Feller process.

7.2 The generator

Motivation: If \(\phi:[0,\iy)\to \R\), additive, \(\phi(0)=1\), then \(\phi(t) = e^{at}\), \(a=\ddd t^{+}\phi(t)|_{t=0}\).

Definition. Let \((P_t)_{t\ge 0}\) be Feller semigroup on \(C_\iy(\R^d)\). Then \[\begin{align} Au :&= \lim_{t\to 0} \fc{P_t u-u}t\\ D(A) :&= \set{u\in C_\iy(\R^d)}{\exists g\in C_\iy(\R^d), \lim_{t\to 0} \ve{\fc{P_tu-u}t-g}_{\iy}=0} \end{align}\]

is the infinitesimal generator of \((P_t)_{t\ge 0}\). \(A\) is a function \(A:D(A)\to C_\iy(\R^d)\).

Ex. for \(BM^d\), \(P_tu(x) = \E^xu(B_t)\), \(A=\rc 2 \De\) on \(C_\iy^2(\R^d)\). Proof: Taylor expansion.

Lemma 7.10. Let \(P_t\) be Feller semigroup with generator \(A\). \(P_t\) and \(\int_0^t P_s\cdot\,ds\) are \(D(A)\to D(A)\). \[\begin{align} \ddd tP_tu &= AP_tu = P_tAu\\ P_t u - u &= A\int_0^t P_su\,ds. \end{align}\]

Proof. Use: \(P_t\) is contraction to get \(\ddd t^+\). Use this and strong continuity to get \(\ddd t^-\). Fubini’s Theorem for part 2.

Corollary 7.11. Let \((P_t)_{t\ge 0}\) be Feller semigroup with generator \(A\).

1. \(D(A)\) is dense in \(C_\iy(\R^d)\).
2. \(A\) is a closed operator.
3. If \((T_t)_{t\ge 0}\) is also Feller with generator \(A\), \(P_t=T_t\).

Proof.

1. \(\ep^{-1}\int_0^\ep P_su\,ds\to u\). (Decay it a little!)
2. If \(u_n\to u\), \(Au_n\to w\), then \(Au=w\).
3. \(\ddd sP_{t-s} T_su = 0\).

Proposition 7.13. Let \(P_t\) be Feller. Then \(\al U_\al\) is conservative, contraction on \(B_b\), positivity preserving, Feller, strongly continuous on \(C_\iy\). Moreover

1. \((U_\al)_{\al>0}\) is resolvent: \(U_\al = (\al \id -A)^{-1}\).
2. Resolvent equation \(U_\al u - U_\be u = (\be-\al) U_\be U_\al u\), \(u\in B_b\).
3. There is 1-to-1 relationship between \((P_t)\) and \((U_\al)\).
4. \((U_\al)\) is sub-Markovian iff \((P_t)\) is sub-Markovian.

(Intuition for (1). \(\iiy e^{(A-\al I) t}u\,dt = (A-\al I)^{-1}\).)

Example 7.14. For Brownian motion \[ U_\al u(x) = \begin{cases} \int \fc{e^{-\sqrt{2\al}y}}{\sqrt{2\al}}u(x+y)\dy,&d=1\\ \int \rc{\pi^{\fc d2}}\pf{\al}{2y^2}^{\fc d4-\rc2} K_{\fc d2-1}(\sqrt{2\al}y) u(x+y)\dy,&d\ge 2. \end{cases} \]

Determining domain of generator:

Theorem. If \((A', D(A'))\) extends \((A,D(A))\), and for any \(u\in D(A)\), \(A'u=u\implies u=0\)< then \((A,D(A))=(A', D(A'))\). (? I don’t get this.)

8 The PDE connection

For many classical PDE problems probability theory yields concrete representation formulae for the solutions in the form of expected values of a Brownian functional. These formulae can be used to get generalized solutions of PDEs (which require less smoothness of the initial/boundary data or the boundary itself) and they are amenable to Monte-Carlo simulations

Lemma 8.1. \(f\in D(\De)\), \(u(t,x):=\E^x f(B_t)\). \(u\) is unique bounded solution of heat equation with initial value \(f\).

Proof. Laplace transform and IbP. Use uniqueness of resolvent operator.