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 t N. Wiener constructed a rigorous mathemati- cal 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 . x 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 par- ticular: 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 pro- cess. 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. ·.