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A Short Tutorial on the Path Integral Formalism for Stochastic Differential

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A Short Tutorial on the Path Integral Formalism for Stochastic Differential A short tutorial on the path integral formalism for stochastic differential equations Dani Mart´ı Group for Neural Theory, Ecole´ Normale Sup´erieure,Paris, France This is a very short and incomplete introduction to the formalism of path integrals for stochas- tic systems. The tutorial is strongly inspired by the very nice introduction by Chow and Buice (2015), which is far more complete than this text and whose reading I highly recommend if you are interested in the subject. Goal: Provide a general formalism to compute moments, correlations, and transition proba- bilities for any stochastic process, regardless of its nature. The formalism should allow for a systematic perturbative treatment. Preliminaries: Moment generating functionals Single variable Consider a single random variable X with probability density function p(x), not necessarily normalized. The n-th moment of X is defined as R xnp(x) dx hXni ≡ : R p(x) dx Instead of computing the moments one by one, we can make use of the moment generating function, defined as Z Z(λ) ≡ eλxp(x) dx: (1) With this definition, moments can be computed by simply taking derivatives of Z(λ) and dividing the result by Z(0): n n 1 d hX i = n Z(λ) : (2) Z(0) dλ λ=0 The expansion of Z(λ) in a power series reads then 1 X λm Z(λ) = Z(0) hXmi: (3) m! m=0 • Example: cumulant-generating function for a Gaussian random variable X with mean a and variance σ2. Plugging the (unnormalized) Gaussian probability density p(x) = exp[−(x − µ)2=(2σ2)] into Eq. (1) gives Z 1 (x − µ)2 Z(λ) = exp − 2 + λx dx; (4) −∞ 2σ which can be evaluated by `completing the square' in the exponent, 1 1 λ2σ2 − (x − µ)2 + λx = − (x − µ − λσ2)2 + λµ + : 2σ2 2σ2 2 1 Carrying out integral over x yields 1 Z(λ) = Z(0) exp λµ + λ2σ2 ; 2 p R 1 −x2=(2σ2) where Z(0) = −∞ e dx = 2πσ is the normalization factor. Once we have Z(λ) we can easily compute the moments using Eq. (2). Thus, the first and second moments of x are d 1 2 2 hXi = exp λa + λ σ = a; dλ 2 λ=0 2 2 d 1 2 2 2 2 hX i = 2 exp λa + λ σ = a + σ : dλ 2 λ=0 The cumulants of the random variable X, denoted by hhXnii, are defined by n n d Z(λ) hhX ii = n ln : (5) dλ Z(0) λ=0 Equivalently, the expansion of ln Z(λ) in power series is 1 X λn ln Z(λ) = ln Z(0) + hhXnii: (6) n! n=1 The relation between cumulants and moments follows from equating the expansion (6) to the logarithm of the expansion (3), 1 1 1 ! X λn X λn X λn hhXnii = ln hXni = ln 1 + hXni ; (7) n! n! n! n=1 n=0 n=1 expanding the right-hand side of Eq. (7), and equating coefficients of equal powers of λ. The first three cumulants are hhXii = hXi, hhX2ii = hX2i − hXi2, and hhX3ii = hX3i − 3hX2ihXi + 2hXi3. • Example: cumulant-generating function for a Gaussian random variable X with mean a and variance σ2. We just compute the logarithm of the moment-generating function found in the previous example, 1 ln Z(λ) = ln Z(0) + λa + λ2σ2 2 and identify the cumulants through relation (6). We have hhXii = a, hhX2ii = σ2, and hhXnii = 0 for n > 2. Multidimensional variables Let X be a random variable having r components X1;X2;:::;Xr, and with joint probability density p(x) ≡ p(x1; x2; : : : ; xr). The moments of X are R xm1 xm2 ··· xmr p(x) dx hXm1 Xm2 ··· Xmr i = 1 2 r ; 1 2 r R p(x) dx 2 where dx ≡ dx1dx2 ··· dxr. The moment-generating function of X is a straightforward gener- alization of the one-dimensional version Z Z(λ) ≡ exp(λTx) p(x) dx; (8) where here λ is an r-dimensional vector and λT x is just the dot product between λ and x. Moments are generated with 1 @m1 @m2 @mr hXm1 Xm2 ··· Xmr i = ··· Z(λ) (9) 1 2 r m1 m2 mr Z(0) @x1 @x2 @xr λ=0 • Example: moment-generating function for a random Gaussian vector X with zero mean and variance Σ = K, X ∼ N (0; K). If we drop, as usual, the normalization factor of the probability density we have Z 1 Z(λ) = exp − xT K−1x + λT x dx: (10) 2 The matrix K is symmetric and therefore diagonalizable (if K happens to be non- symmetric, the contribution from its antisymmetric part to the quadratic form xT K−1x vanishes). Let us denote by fuig the basis of unit eigenvectors of K, and by U the matrix that results from arranging the set fuig in columns. Because fug is an orthonormal set, UT U = 1. The diagonalization condition can be summarized by KU = UD, where D is 2 the diagonal matrix with the eigenvalue σi at its i-th diagonal entry. In terms of the new −1 −1 T P P basis, K = UD U , x = i ciui = Uc, and λ = i diui = Ud, where c and d are coefficient vectors. We have 1 1 Y c2 exp − xT K−1x + λTx = exp − cT D−1c + dTc = exp − i + c d ; 2 2 2σ2 i i i i and therefore Z Y c2 Y Z c2 Z(λ) = exp − i + c d jUj dc; = exp − i + c d dc 2σ2 i i 2σ2 i i i i i i i 2 2 Y 1=2 d σ = 2πσ2 exp i i = j2πDj1=2 expdT Dd i 2 i = j2πKj1=2 expλT Kλ: (11) where in the first line we used that the Jacobian jUj is 1 for orthogonal transformations, and where in the second line we completed the square for all the decoupled factors. Although the Gaussian example is just a particular case, it is an important one because it can be computed in a closed form and because it is the basis for many perturbative schemes. The moment generating function Z(λ) is an exponential of a quadratic form in λ, which implies that only moments of even order will survive (see Eq. (9)). In particular we have 1 @ @ hxaxbi = Z(λ) = Kab; Z(0) @xa @xb λ=0 where a; b are any two indices of the components of x. 3 Moreover, it is not difficult to convince oneself that for Gaussian vectors any moment of even-order can be obtained as a sum of all the possible pairings between variables, for example: hxaxbxcxdi = hxaxbihxcxdi + hxaxcihxbxdi + hxaxdihxbxci: This property is usually referred to as Wick's theorem, or Isserlis's theorem. Fields We can extend from finite-dimensional systems to functions defined on some domain [0;T ]. The basic idea is to discretize the domain interval into n segments of length h such that T = nh, and then take the limits n ! 1 and h ! 0 while keeping T = nh constant. Under this limit the originally discrete set ft0; t1; : : : ; tn = T g becomes dense on the interval [0;T ]. Any t can then be approximated as t = jh, for some j = 0; 1; : : : ; n and we can identify xj −! x(t); λj −! λ(t);Kij −! K(s; t): Now instead of a moment-generating function we'll have a moment-generating functional, an object that maps functions into real numbers. For the Gaussian case we have the functional version of Eqs. (10) and (11) Z 1 ZZ Z Z[λ(t)] = Dx(s) exp − x(u)K−1(u; v)x(v) du dv + λ(u)x(u) du 2 1 ZZ = Z[0] exp λ(u)K(u; v)λ(v) du dv (12) 2 where we use square brackets to denote a functional, and where Z[0] is the value of the generating functional with the external source λ(t) set to 0. We also defined the measure for integration over functions as n Y Dx(t) ≡ lim dxi: n!1 i=0 The integral in Eq. (12) is an example of path integral, or functional integral (Cartier and DeWitt-Morette, 2010). The presentation here is obviously very handwavy, and there are several ill-defined quan- 1=2 tities lying around like, for instance, the prefactor Z[0] = limn!1 j2πKj , which is formally infinite. Fortunately, the moments we derive from the moment-generating functional are well- defined. Now we need to compute the moments, for which we need the notion of functional deriva- tive. Although functional derivatives can be defined rigorously, here we only need to remember the following axioms δλ(t) @λj = δ(t − s) equivalent to = δij ; δλ(s) @λi ! δ Z @ X λ(s)x(s) ds = x(t) equivalent to λ x = x ; δλ(t) @λ j j i i j where δ(·) is the Dirac delta, or point-mass functional, and δij is the Kronecker delta: δij = 1 if i = j, 0 otherwise. As usual, moments are obtained from taking functional derivatives on the moment-generating functional: * + Y 1 Y δ x(ti) = Z[λ] : Z[0] δλ(ti) i i λ(t)=0 4 • Example: given the moment-generating functional of a Gaussian process, Eq. (12), the two-point correlation function is 1 δ δ hx(t)x(s)i = Z[λ] = K(t; s): Z[0] δλ(t) δλ(s) λ(u)=0 More generally, any (even-order) moment will be of the form * + Y 1 Y δ X x(t ) = Z[λ] = K(t ; t ) ··· K(t ; t ); i i1 i2 i2s−1 ti2s Z[0] δλ(ti) i i λ(t)=0 all possible pairings because of Wick's theorem.
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