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SRQ seminar – September 17th, 2018

Notes from a talk in the SRQ series.

INI Seminar, 20180917. Weber.

Stochastic dynamics. Summary.

  1. Connection between field theory, measures \(e^{- V}\), & SPDEs

  2. Scaling of SPDEs, subcriticality

  3. Function spaces

1Langevin dynamics

\(V : \mathbb{R}^d \rightarrow \mathbb{R}_+\) sufficient growth, \(C^{\infty}\). Measure \(\mu = e^{- V (x)} \mathrm{d} x / Z\) on \(\mathbb{R}^d\). How to sample from this measure?

One possible dynamics: Langevin SDE.

\(\displaystyle \mathrm{d} x_t = - \nabla V (x_t) \mathrm{d} t + \sqrt{2} \mathrm{d} w_t, \quad t \geqslant 0.\)

Theorem 1. This SDE defines a reversible Markov process for \(\mu\).

This means that if \(x_0 \sim \mu\) and \(\mu \bot w\) then \((x_t)_{t \in [0, 1]}\) has the same law as \((x_{1 - t})_{t \in [0, 1]}\) (the same process run backwards from \(t = 1\)). In particular \(x_1\) has the same law as \(x_0\), that is \(\mu\). The measure \(\mu\) is therefore invariant under this Markovian evolution.

Proof. Check detailed balance condition. Namely check that the generator \(\mathcal{L}\) of the SDE is symmetric wrt. \(\mu\).

\(\displaystyle \int f (x) \mathcal{L}g (x) \mu (\mathrm{d} x) = \int \mathcal{L}f (x) g (x) \mu (\mathrm{d} x)\)

with \(\mathcal{L}= \Delta - \nabla V \cdot \nabla\). Explicitly (assume \(Z = 1\))

\(\displaystyle \int f (x) \mathcal{L}g (x) \mu (\mathrm{d} x) = \int f (x) \Delta g (x) e^{- V (x)} \mathrm{d} x - \int f (x) \nabla V \cdot \nabla g (x) e^{- V (x)} \mathrm{d} x\)

(integrating by parts the Laplacian)

\(\displaystyle = \int \nabla g (x) \cdot \nabla [f (x) e^{- V (x)}] \mathrm{d} x - \int f (x) \nabla V \cdot \nabla g (x) e^{- V (x)} \mathrm{d} x = \int (\nabla g (x) \cdot \nabla f (x)) e^{- V (x)} \mathrm{d} x\)

which is symmetric in \(f, g\) and we can undo the same computation on the \(f\) side. \(\Box\)

This computation remains valid for modifications of the SDE of the form

\(\displaystyle \mathrm{d} x_t = - AA^t \nabla V (x_t) \mathrm{d} t + \sqrt{2} A \mathrm{d} w_t, \quad t \geqslant 0.\)

where \(A \in \mathcal{L} (\mathbb{R}^m ; \mathbb{R}^d)\) for any \(m \geqslant d\). By choosing appropriately \(A\) we can generate both Glauber and Kawasaki dynamics.

Both gradient and Brownian motion are defined wrt. a quadratic form. For BM we mean that if \(v, w \in \mathbb{R}^d\) there is a quadratic form \(Q\) associated to the covariance of the Brownian motion as:

\(\displaystyle \mathbb{E} (B_t \cdot v) (B_s \cdot w) = (t \wedge s) Q (v, w) .\)

Aim: understand the ultraviolet problem from the construction of \(\phi^4\). Consider a lattice with spacing \(N^{- 1}\)

\(\displaystyle \Lambda_N = (N^{- 1} \mathbb{Z}/\mathbb{Z})^d,\)

and a scalar field \(\phi : \Lambda_N \rightarrow \mathbb{R}\) with potential

\(\displaystyle V_N (\phi) = \frac{1}{N^d} \sum_{x \in \Lambda_N} \left[ F (\phi (x)) + \underbrace{\frac{1}{2} | \nabla \phi (x) |^2}_{\frac{N^2}{2} \sum_{x \sim y} | \phi (x) - \phi (y) |^2} \right],\)

with \(F\) is nice and sufficiently growning.

Let us compute the Langevin (gradient) dynamics.

We have to fix a scalar product and we start with the \(L^2 (\Lambda_N)\) scalar product

\(\displaystyle \langle \psi, \varphi \rangle_{L^2} = \frac{1}{N^d} \sum_{x \in \Lambda_N} \psi (x) \varphi (x)\)

The gradient \(\nabla V\) has the form

\(\displaystyle \left. \frac{\mathrm{d}}{\mathrm{d} \varepsilon} \right|_{\varepsilon = 0} V_N (\phi + \varepsilon e) = \frac{1}{N^d} \sum_{x \in \Lambda_N} F' (\phi (x)) e (x) + \frac{1}{N^d} \sum_{x \in \Lambda_N} \nabla \phi (x) \cdot \nabla e (x) \)

and an integration by parts gives

\(\displaystyle = \langle F' (\phi) - \Delta \phi, e \rangle_{L^2}\)

so can identify the gradient \(\nabla V\) with the l.h.s. of this scalar product and let

\(\displaystyle \nabla V (\phi) (x) = F' (\phi (x)) - \Delta \phi (x) .\)

Form of the noise: we want a noise \(W_t\) such that

\(\displaystyle \mathbb{E} [\langle W_t, \psi_1 \rangle_{L^2} \langle W_t, \psi_2 \rangle_{L^2}] = (t \wedge s) \langle \psi_1, \psi_2 \rangle_{L^2}\)
\(\displaystyle =\mathbb{E} \left[ \frac{1}{N^d} \sum_{x_1 \in \Lambda_N} \psi_1 (x_1) W_t (x_1) \frac{1}{N^d} \sum_{x_2 \in \Lambda_N} \psi (x_2) W_s (x_2) \right] = (t \wedge s) \frac{1}{N^d} \sum_{x \in \Lambda_N} \psi_1 (x) \psi_2 (x)\)

so we need to take

\(\displaystyle \mathbb{E} [W_t (x_1) W_s (x_2)] = (t \wedge s) N^d \delta_{x_1, x_2} .\) (1)

This suggests that the limiting equation as \(N \rightarrow \infty\) should be

\(\displaystyle \mathrm{d} \phi_t = (\Delta \phi - F' (\phi)) \mathrm{d} t + \sqrt{2} \mathrm{d} W_t\)

where \((W_t)_t\) is a Brownian motion with covariance “\(\mathbb{E} [W_t (x) W_t (y)] = \delta_{\mathbb{T}^d} (x - y)\)”. This dynamics is usually called: Allen–Cahn, or Glauber or Model A.

Another possible choice is to consider a Kawasaki–like dynamics

\(\displaystyle \mathrm{d} \phi_t = (- \Delta) (\Delta \phi - F' (\phi)) \mathrm{d} t + \sqrt{2} (- \Delta)^{1 / 2} \mathrm{d} W_t\)

which is called the Cahn–Hillard equation (or Kawasaki, or Model B). It is an equation which preserves the averages and can be cast in (stochastic) conservation law form.

Reversibility: heavily used in stochastic quantisation (Albeverio, Roeckner) to understand solutions in \(2 d\). More recent techniques for dealing with singular equations (regularity structures and company) ignore reversibility. E.g.

\(\displaystyle \mathrm{d} \phi_t = (\Delta \phi - f (\phi)) \mathrm{d} t + \sqrt{2} \mathrm{d} W_t, \qquad \phi : \Lambda \rightarrow \mathbb{R}^2\)

and \(f\) does not need to be a gradient, e.g. a polynomial with some coercivity properties.

(Jona–Lasinio, Séneor have analysed non-gradient perturbations of gradient dynamics using Girsanov formula).

2Scaling

2.1Scaling of white noise

White noise \(\xi\) want to be defined as the Gaussian process with covariance given by Dirac–\(\delta\). It makes sense to take it as a random distribution in \(\mathcal{D}' (\mathbb{R}_+ \times \mathbb{R}^d)\) with covariance

\(\displaystyle \mathbb{E} [\xi (\eta)^2] = \| \eta \|_{L^2 (\mathbb{R}_+ \times \mathbb{R}^d)}, \qquad \eta \in \mathcal{D} (\mathbb{R}_+ \times \mathbb{R}^d) . \) (2)

Realization: On \(\mathbb{R}_+\) take \((B_t)_{t \geqslant 0}\) be a Brownian motion, define \(\xi (\eta) = \int_{\mathbb{R}_+} \eta (t) \mathrm{d} B_t\) and then  (2) is just Ito¯ isometry.

On \(\mathbb{T}^d\) define \(\xi\) via Fourier series

\(\displaystyle \xi (x) = \sum_{k \in \mathbb{Z}^d} e^{i k \cdot x} G_k\)

where \((G_k)_{k \in \mathbb{Z}^d}\) are centered Gaussian, complex valued with \(G_k = (G^1_k + i G^2_k) / \sqrt{2}\) with \(G^i_k \in \mathbb{R}\) and \(G^1, G^2 \sim \mathcal{N} (0, 1)\) and independent up to \(G_k = G_{- k}^{\ast}\) and the structure of the covariance follows from independence.

Cylindrical Brownian can be seen as a superposition of independent Brownian motions in different Fourier modes.

Scaling for Brownian motion: \(\lambda^{- 1 / 2} B_{\lambda t} \sim B_t\). Define \(\xi (\alpha t, \beta x) = \xi_{\alpha, \beta}\) via \(\xi_{\alpha, \beta} (\eta) = \xi (S_{\alpha, \beta} \eta)\) where

\(\displaystyle S_{\alpha, \beta} \eta (t, x) = \alpha^{- 1} \beta^{- d} \eta (t / \alpha, x / \beta)\)

where this scaling preserves the \(L^1\) norm of the test function.

\(\displaystyle \mathbb{E} [\xi_{\alpha, \beta} (\eta)^2] =\mathbb{E} [\xi (S_{\alpha, \beta} \eta)^2] = \| S_{\alpha, \beta} \eta \|_{L^2}^2 = \int_{\mathbb{R}_+ \times \mathbb{R}^d} | \alpha^{- 1} \beta^{- d} \eta (t / \alpha, x / \beta) |^2 \mathrm{d} t \mathrm{d} x = \alpha^{- 1} \beta^{- d} \| \eta \|_{L^2}^2\)

so

Corollary 2. For all \(\alpha, \beta > 0\),

\(\displaystyle \alpha^{1 / 2} \beta^{d / 2} \xi_{\alpha, \beta} \sim \xi .\)

Informally

\(\displaystyle \alpha^{1 / 2} \beta^{d / 2} \xi (\alpha t, \beta x) \xequal{\text{law}} \xi (t, x) .\)

2.2Scaling for SPDEs, linear case

Let us start from linear equations

\(\displaystyle (\partial_t - \Delta) Z = \xi .\)

Rescale \(\hat{Z} (t, x) := \lambda^{\delta} Z (\lambda^{\alpha} t, \lambda^{\beta} x)\). \(\hat{\xi} (t, x) := \lambda^{(\alpha + d \beta) / 2} \xi (\lambda^{\alpha} t, \lambda^{\beta} x)\) and \(\hat{\xi} \sim \xi\). Now

\(\displaystyle \partial_t \hat{Z} (t, x) = \lambda^{\delta} \lambda^{\alpha} Z (\lambda^{\alpha} t, \lambda^{\beta} x), \qquad \Delta \hat{Z} (t, x) = \lambda^{\delta} \lambda^{2 \beta} Z (\lambda^{\alpha} t, \lambda^{\beta} x) .\)

So we need to take \(\alpha = 2 \beta\) and \((\alpha + d \beta) / 2 = \alpha + \delta\) so \(\alpha = 2 \beta\) and \(\delta = (d \beta - \alpha) / 2 = \beta (d - 2) / 2\). Therefore

\(\displaystyle \lambda^{- (d / 2 - 1)} Z (\lambda^2 t, \lambda x) \xequal{\text{law}} Z (t, x) .\)

The exponent in the prefactor “hints” to the regularity of the random field \(Z\).

2.3Scaling for SPDEs, nonlinear equations

If we are interested in the \(\phi^4\) theory, that is the case where \(V (\phi) = \frac{1}{4} \int \phi (x)^4 + \frac{1}{2} | \nabla \phi (x) |^2\) then the equation is

\(\displaystyle (\partial_t - \Delta) \phi = - \phi^3 + \xi .\)

Perform rescaling \(\hat{\phi} (t, x) = \lambda^{(d / 2 - 1)} \phi (\lambda^2 t, \lambda x)\), the one suggested by the linear equation. Then the equation in the new variables reads

\(\displaystyle (\partial_t - \Delta) \hat{\phi} = - \lambda^{4 - d} \hat{\phi}^3 + \hat{\xi} .\)

For the UV problem, we need to think about \(\lambda\) being smaller and smaller (why?) and this makes the non-linear term small in this regime. So for \(d < 4\) we are expecting that in small scales \((\lambda \rightarrow 0)\) we are expecting that the stochastic heat equation (the linear equation behaviour) dominates.

Idea: We try to build a perturbative expansion, on small scales, around the linear theory. At some point one can stop and analyse the remainder with PDE methods.

In regularity structure there exists results which allow to constuct and solve for small times a large class of equations in the subcritical regime.

For \(\phi^4\) the long time and large space problem is well understood by now.