18.175: Lecture 37 More Brownian motion Scott Sheffield MIT 1 18.175 - PowerPoint PPT Presentation

18.175: Lecture 37 More Brownian motion Scott Sheffield MIT 1 18.175 Lecture 37

Outline Brownian motion properties and construction Markov property, Blumenthal’s 0-1 law 2 18.175 Lecture 37

Outline Brownian motion properties and construction Markov property, Blumenthal’s 0-1 law 3 18.175 Lecture 37

Basic properties � Brownian motion is real-valued process B t , t ≥ 0. � Independent increments: If t 0 < t 1 < t 2 . . . then B ( t 0 ) , B ( t 1 − t 0 ) , B ( t 2 − t 1 ) , . . . are independent. � Gaussian increments: If s , t ≥ 0 then B ( s + t ) − B ( s ) is normal with variance t . � Continuity: With probability one, t → B t is continuous. � Hmm... does this mean we need to use a σ -algebra in which the event “ B t is continuous” is a measurable? � Suppose Ω is set of all functions of t , and we use smallest σ -field that makes each B t a measurable random variable... does that fail? 4 18.175 Lecture 37

Basic properties Translation invariance: is B t 0 + t − B t 0 a Brownian motion? � � Brownian scaling: fix c , then B ct agrees in law with c 1 / 2 B t . � � Another characterization: B is jointly Gaussian, EB s = 0, � � EB s B t = s ∧ t , and t → B t a.s. continuous. 5 18.175 Lecture 37

Defining Brownian motion Can define joint law of B t values for any finite collection of � � values. Can observe consistency and extend to countable set by � � Kolmogorov. This gives us measure in σ -field F 0 generated by cylinder sets. But not enough to get a.s. continuity. � � Can define Brownian motion jointly on diadic rationals pretty � � easily. And claim that this a.s. extends to continuous path in unique way. We can use the Kolmogorov continuity theorem (next slide). � � Can prove H¨ older continuity using similar estimates (see � � problem set). Can extend to higher dimensions: make each coordinate � � independent Brownian motion. 6 18.175 Lecture 37

Continuity theorem Kolmogorov continuity theorem: Suppose � � E | X s − X t | β ≤ K | t − s | 1+ α where α, β > 0. If γ < α/β then with probability one there is a constant C ( ω ) so that | X ( q ) − X ( r ) | ≤ C | q − r | γ for all q , r ∈ Q 2 ∩ [0 , 1]. Proof idea: First look at values at all multiples of 2 − 0 , then � � at all multiples of 2 − 1 , then multiples of 2 − 2 , etc. At each stage we can draw a nice piecewise linear � � approximation of the process. How much does the approximation change in supremum norm (or some other H¨ older norm) on the i th step? Can we say it probably doesn’t change very much? Can we say the sequence of approximations is a.s. Cauchy in the appropriate normed spaced? 7 18.175 Lecture 37

Continuity theorem proof Kolmogorov continuity theorem: Suppose � � E | X s − X t | β ≤ K | t − s | 1+ α where α, β > 0. If γ < α/β then with probability one there is a constant C ( ω ) so that | X ( q ) − X ( r ) | ≤ C | q − r | γ for all q , r ∈ Q 2 ∩ [0 , 1]. Argument from Durrett (Pemantle): Write � � G n = {| X ( i / 2 n ) − X (( i − 1) / 2 n ) |} ≤ C | q − r | λ for 0 < i ≤ 2 n } . Chebyshev implies P ( | Y | > a ) ≤ a − β E | Y | β , so if � � λ = α − βγ > 0 then P ( G c ) ≤ 2 n · 2 n βγ · E | X ( j 2 − n ) | β = K 2 − n λ . n 8 18.175 Lecture 37

Easy observations Brownian motion is H¨ older continuous for any γ < 1 / 2 (apply � � theorem with β = 2 m , α = m − 1). Brownian motion is almost surely not differentiable. � � Brownian motion is almost surely not Lipschitz. � � Kolmogorov-Centsov theorem applies to higher dimensions � � (with adjusted exponents). One can construct a.s. continuous functions from R n to R . 9 18.175 Lecture 37

Outline Brownian motion properties and construction Markov property, Blumenthal’s 0-1 law 10 18.175 Lecture 37

Outline Brownian motion properties and construction Markov property, Blumenthal’s 0-1 law 11 18.175 Lecture 37

More σ -algebra thoughts Write F o = σ ( B r : r ≤ s ). � � s Write F = ∩ t > s F o + � � s t + + Note right continuity: ∩ t > s F t = F s . � � + F allows an “infinitesimal peek at future” � � s 12 18.175 Lecture 37

Markov property If s ≥ 0 and Y is bounded and C -measurable, then for all � � x ∈ R d , we have E x ( Y ◦ θ s |F + ) = E B s Y , s where the RHS is function φ ( x ) = E x Y evaluated at x = B s . Proof idea: First establish this for some simple functions Y � � (depending on finitely many time values) and then use measure theory (monotone class theorem) to extend to general case. 13 18.175 Lecture 37

Looking ahead Theorem: If Z is bounded, measurable then for s ≥ 0 have � � � E x ( A |F + ) = E x ( Z |F 0 ) . s s 14 18.175 Lecture 37

Blumenthal’s 0-1 law If A ∈ F + , then P ( A ) ∈ { 0 , 1 } (if P is probability law for � � 0 Brownian motion started at fixed value x at time 0). There’s nothing you can learn from infinitesimal neighborhood � � of future. 15 18.175 Lecture 37

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