1 Introduction. 2 White noise. The content of these notes is also covered by chapter 4 section A of [1].

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1 1 Introduction The content of these notes is also covered by chapter 4 section A of [1]. 2 White noise In applications infinitesimal Wiener increments are represented as dw t = η t The stochastic process η t is referred to as white noise. Consistence of the definition requires η t to be a Gaussian process with the following properties. Zero average: Namely we must have δ-correlation: at any instant of time as it follows from the identification η t = = w t+ w t = η t η t η s = σ 2 δ (1) (t s), σ 2 > ds w tw s = η t η s (2.1) Namely, writing the correlation of the Wiener process in terms of the Heaviside step function and observing we obtain, upon inserting (2.2) into (2.1) w t w s = σ 2 [s H(t s) + t H(s t)] (2.2) d H(t s) = δ(1) (t s) ds w tw s = σ 2 t [H(t s) s δ(1) (t s)] + σ 2 s [H(s t) t δ(1) (s t)] By construction the δ is even an even function of its argument: the right hand side can be couched into the form [ ds w tw s = σ 2 2 δ (1) (t s) + (t s) d ] δ(1) (t s) In order to interpret the meaning of the derivative, we integrate the right hand side over a smooth test function f s+ε f(t) (t s) t δ(1) (t s) = = σ+ε σ+ε df (t) (t s) δ(1) (t s) σ+ε f(t) δ (1) (t s) f(t) δ (1) (t s) = f(s) (2.3) 1

2 We conclude that ds w tw s = σ 2 δ (1) (t s) the identity determining the value of the white noise correlation. 3 Paley-Wiener-Zygmund integral Definition 3.1 (Paley-Wiener-Zygmund integral). Let g : [, T ] R a continuously differentiable function such that g() = g(t ) = The random variable is defined as G T = dw t g(t) dg dw t g(t) = w t (t) The Paley-Wiener-Zygmund integral can be tackled resorting to standard Lebesgue-Stieltjes integration theory. We can prove Proposition 3.1. i G T = Proof. ii G 2 T = i it follows by exchanging the order between integral and average. ii By definition we have G 2 T = If we now use the calculation of section (2), we get into G 2 T = ds dg (t)dg ds (s) w tw s = ds g(t)g(s) d2 ds w tw s (for σ 2 = 1) which proves the claim. ds g(t)g(s) δ (1) (t s) = g 2 (t) 2

3 4 Gaussian statistics and δ-correlation Gaussian statistics and δ-correlation imply that η t is independent of η s for any t s. The claim is verified by inspection of the characteristic function of the white-noise. The Paley-Wiener-Zygmund integral allows us to write for some smooth g chosen as in section 3 e ıλ ηtg(t) e ıλ dwtg(t) = e ıλ wtg (t) The leftmost integral can be computed using e.g. the Karhunen-Loève representation of the Brownian motion here we used the shorthand notation e ıλ wtg (t) = e ıλ P n cn ψn(t) g (t) g := dg Randomness is stored in the {c n } n= which form a sequence of independent Gaussian random variables with zero average and variance for c n equal to the n-th eigenvalue of the operator defined by Thus we obtain R(t, s) = w t w s e ıλ wtg (t) = e λ2 P 2 n= ds rn ψn(t)ψn(s) g (t)g (s) = e λ2 2 since by construction The lemma in 3 then guarantees us We can read the result in two ways. Characteristic function of R(t, s) = r n ψ n (t)ψ n (s) n= e ıλ ηtg(t) e ıλ dwtg(t) = e λ2 2 ds R(t,s) g (t)g (s) We have just proved that G T := dw t g(t) e ı λ G T = e λ2 2 G2 T i.e. that G T has a Gaussian distribution. Characteristic function of the white noise. Let us set λ equal to unit and inspect e ı ηt g(t) = e 1 2 (4.1) 3

4 Interpreting integrals as sums η t g(t) k η tk g(t k ) were the η tk a collection of random Gaussian variables with correlation η tk η tl = C kl we would write e ı P k ηt k g(t k) = e 1 2 P k P l g(t k)c k l g(t l ) Contrasting this latter result with (4.1) it is tempting to identify k l g(t k )C k l g(t l ) ds g(t)c(t s) g(s) and C(t s) = δ (1) (t s) This is in agreement with the claim that white noise is Gaussian with zero average and δ-dirac covariance. From (4.1) we can also read all the moments of the white noise. In order to do so we need to take functional derivatives with respect to g(t) (see appendix A, in practice treat the function argument as an index and replace δ-kroenecker with δ-functions). For any < s < T implies δ R δg(s) T eı g(t)ηt = δ δg(s) e 1 2 = e 1 2 g(t) δ (1) (t s) = g(s)e 1 2 η s = Furthermore for < s u < T implies δ 2 R δg(s)δg(u) T eı g(t)ηt = δ δg(u) g(s) e 1 2 = δ (1) (u s) e g(s) g(u) e 1 2 η s η u = δ (1) (u s) which recovers for σ 2 = 1 the result obtained in section 2 4

5 A Functional derivative (for practical purposes) Consider a functional space of continuous/smooth functions φ (eventually also satisfying certain boundary conditions) and a functional F [φ]. We define the functional derivative of F, denoted δf/δφ(x), as the distribution such that for all test functions f: x R d δφ(x) f(x) := d F [φ + ε f] dε ɛ= Thus if the definition yields Alternatively, we can define with ϕ x specified by R d x F [φ] = x φ 2 (x) R d δφ(x) f(x) = 2 x φ(x)f(x) R d δφ(x) = lim F [ϕ x ] F [φ] ε ε ϕ x (y) = φ(y) + ε δ (d) (x y) (A.1) For the the example (A.1) this means δφ(x) = 2 φ(x) References [1] L.C. Evans, An Introduction to Stochastic Differential Equations, lecture notes, edu/ evans/ 5

1 Introduction. 2 Diffusion equation and central limit theorem. The content of these notes is also covered by chapter 3 section B of [1].

1 Introduction. 2 Diffusion equation and central limit theorem. The content of these notes is also covered by chapter 3 section B of [1]. 1 Introduction The content of these notes is also covered by chapter 3 section B of [1]. Diffusion equation and central limit theorem Consider a sequence {ξ i } i=1 i.i.d. ξ i = d ξ with ξ : Ω { Dx, 0,