Brownian Motion and the Heat Equation

Brownian motion and the heat equation

Denis Bell

University of North Florida 1. The heat equation

Let the function u(t, x) denote the temperature in a rod at position x and time t

u(t,x)

Then u(t, x) satisfies the heat equation ∂u 1∂2u = , t > 0. (1) ∂t 2∂x2 It is easy to check that the Gaussian function 1 x2 u(t, x) = e−2t 2πt satisfies (1). Let φ be any! bounded continuous function and define

1 (x y)2 u(t, x) = ∞ φ(y)e− 2−t dy. 2πt "−∞ Then u satisfies!(1). Furthermore making the substitution z = (x y)/√t in the integral gives − 1 z2 u(t, x) = ∞ φ(x y√t)e−2 dz − 2π "−∞ ! 1 z2 φ(x) ∞ e−2 dz = φ(x) → 2π "−∞ ! as t 0. Thus ↓ 1 (x y)2 u(t, x) = ∞ φ(y)e− 2−t dy. 2πt "−∞ ! = E[φ(Xt)] where Xt is a N(x, t) random variable solves the heat equation

∂u 1∂2u = ∂t 2∂x2 with initial condition u(0, ) = φ. ·

Note: The function u(t, x) is smooth in x for t > 0 even if is only continuous. 2. Brownian motion In the nineteenth century, the botanist Robert Brown observed that a pollen particle suspended in liquid undergoes a strange erratic motion (caused by bombardment by molecules of the liquid)

Letting w(t) denote the position of the particle in a fixed direction, the paths w typically look like this

N. Wiener constructed a rigorous mathematical model of Brownian motion in the 1930s. The mathematical model of Brownian motion (Wiener process) satisfies the following axioms:

(i) w = 0 and the paths t w are continuous 0 &→ t a.s.

(ii) The increments w ws are independent of t − wu/ u s for all t > s. { ≤ }

(iii) w ws has a normal distribution with mean t− 0 and variance t s. −

x In particular, if we define wt = wt + x (Wiener process started at x) then by (iii) wt has a x N(0, t) distribution. Hence wt has a N(x, t) x distribution, so u(t, x) = φ(wt ) solves the heat equation. More generally, let w= (w1, . . . , wn) consist of n independent copies of 1-dimensional Wiener process. Then for any bounded continuous function Φ : Rn R, u(t, x) E[φ(wx)] solves &→ ≡ t the n-dimensional heat equation ∂u 1 = ∆u ∂t 2 with initial condition Φ, where ∆ is the Lapla- cian n ∂2 ∆ = . 2 ∂x i#=1 i

The Wiener process has many intriguing properies. In particular

1) With probability 1, the path t w is non- &→ t differentiable almost everywhere (wrt Lebesgue measure).

2) w is a Markov process. { t} 3. Markov Processes

Definition. Say the stochastic process ξ , t { t ≥ 0 is a (time-homogeneous) Markov process if } the conditional distribution of ξt+s given

ξu, u s, ξs = x { ≤ } x is the same as the distribution of ξt .

s t+s

Define the associated semigroup P , t 0 { t ≥ } (P φ)(x) E[φ(ξx)]. t ≡ t acting on bounded continuous functions φ. Theorem. P = Ps P . t+s ◦ t

Proof. x Pt+sφ(x) = E[φ(ξs+t)]

= E E[φ(ξx )/ ξx, u s] s+t u ≤ $ % x = E[(Ptφ)(ξs )] = P s(Ptφ)(x).

Define the (infinitesimal) generator of ξ P φ φ Aφ = lim t − . t 0 t ↓ Theorem. For φ in the domain of A, the func- x tion u(t, x) (Ptφ)(x) (= E[φ(ξt )]) solves the Cauchy problem≡ ∂u = Au ∂t u(0, ) = φ. · Proof. Note x u(0, x) = E[φ(ξ0)] = E[φ(x)] = φ(x).

Furthermore, since u = Ptφ, we have

∂u P φ Ptφ = lim t+h − ∂t h 0 h ↓ P (P φ) P φ = lim h t − t h 0 h ↓

= A(Ptφ)

= Au. 4. The Ito integral

The paths of the Wiener process w are of un- bounded variation on every time interval [0, T ]. Nevertheless the Ito (stochastic) integral T f(s)dws "0 can be shown to exist as the limit in probability of the sequence of Riemann type sums n 1 T − f(s)dws = lim f(ti)[w(ti+1) w(ti)] 0 − " i#=1 with the limit taken over a sequence of parti- tions 0 = t0 < t1 < < tn = T of [0, T ] with mesh {tending to zero.· · · } In order to show this limit exists, we must as- sume that f satisfies some conditions. In particular: f = f(w) is non-anticipating i.e. f(t) depends only on wu, u t . { ≤ } Properties of the Ito integral:

1) If T [Ef 2(s)]ds < then 0 ∞ & T E fdw = 0. ' "0 (

The Ito integral does not satisfy the chain rule of classical calculus. It satisfies, instead, a re- markable second-order chain rule, known as:

t 2) Ito s formula: Let ξt = ξ0 + 0 fdw and write this as dξ = fdw. If φ is a C&2 function then η φ(ξ ) satisfies t ≡ t 1 dη = φ (η )dξ + φ (η )f 2(t)dt. t + t t 2 ++ t

We remarked earlier that w is a Markov process. What is its generator? - We can com- pute it using the above properties. Note that t x wt = x + 1dw. "0 Applying Ito s formula with f = 1, we have 1 dφ(wx) = φ (wx)dw + φ (wx)dt. t + t 2 ++ t Writing this in integral form t t x x 1 x φ(wt ) = φ(x) + φ+(ws )dw + φ++(ws )ds. "0 2 "0 Taking expectation of each side, using 1) above, differentiating wrt t and setting t = 0, we have

d x Aφ(x) = E[φ(wt )] dt)t=0 t 1 d x = E[φ++(ws )]ds 2dt)t=0 "0 1 = φ (x). 2 ++ In view of the first part of the talk, this gives x another proof that u(t, x) = E[φ(wt )] satisfies ∂u 1∂2u = . ∂t 2∂x2

This is a very complicated proof of the result, which we derived earlier simply from the fact x that wt has a N(x, t) distribution.

However, the stochastic method is valid in a much more general setting. 5. A generalized heat equation Let a and b be Lipschitz functions. Then it can be shown that the stochastic differential equation t t x x x ξt = x + a(ξs )dw + b(ξs )ds "0 "0 has a unique continuous non-anticipating solu- tion ξ for given initial point x.

Furthermore, ξ is a Markov process. To see this, write for s < t t t x x x x ξt = ξs + a(ξs )dw + b(ξs )ds. "s "s A similar calculation to above shows that the generator of ξ is the differential operator 1 Aφ(x) = a2(x)φ (x) + b(x)φ (x). 2 ++ + Thus the function u(t, x) E[φ(ξx)] solves ≡ t ∂u = Au ∂t with initial condition u(0, ) = φ. ·