# Is there an infinite dimensional Stein's lemma?

Classical Stein's lemma says that if $$\mathbf{X}$$ is a centered Gaussian random vector and $$g$$ is a function which is nice enough, we have $$\mathbb{E} \, X_i \, g ( \mathbf{X} ) = \sum_k \operatorname{cov} (X_i, X_k)\, \mathbb{E} \frac{\partial g}{\partial x_k}.$$ I wonder if there is some infinite dimensional analogue of this lemma, saying something like $$\mathbb{E} \, X ( t ) \, g ( X) = \int \mathbb{E} \operatorname{cov}(X(t), X(s)) D_{X(s)} g \ ds$$ if $$X$$ is a continuous real-valued Gaussian process satisfying some assumptions, $$g$$ is some nice enough functional and $$D$$ is some derivative-like operator.

I know that my question is rather vague, I was curious since I needed to compute expectations like the one on the left and Stein's lemma immediately popped in mind.

Mostly unrelated question. Most of the sources demand that $$g$$ be continuously differentiable for the classical lemma to hold. It seems that this is not necessary. It should be enough for these derivatives $$\partial_k g$$ to exist in the distributional sense, because they are immediately integrated against normal density. Is there some source where the lemma is formulated like this?

• Should $\mathbb{E} \, X_i \, g ( \mathbf{X} ) = \mathbb{E} \, (X_i-\mu_i) \, g ( \mathbf{X} )$ or does that not matter? Wikipedia uses the latter definition: en.wikipedia.org/wiki/Stein%27s_lemma Commented Nov 7, 2023 at 17:10
• @SidharthGhoshal, I just forgot the word "centered", thanks for the remark. Commented Nov 7, 2023 at 17:19
• Assuming that $g(X)$ can be approximated well enough by smooth enough functions of restrictions of the random function $X$ to finite sets, you can use the "finite-dimensional" Stein lemma. Commented Nov 7, 2023 at 18:50
• Yes the integration by parts formula in Malliavin calculus eg. in "Stein's method on Wiener chaos" (arxiv.org/pdf/0712.2940.pdf) eq. (1.23). Commented Nov 7, 2023 at 18:55
• Theorem 5.1.8 on page 209 of the book "Gaussian Measures" by Bogachev provides a generalization to possibly infinite dimensional spaces. Commented Nov 7, 2023 at 19:48

It seems that I have found what I needed under the name of Novikov-Furutsu theorem in the following papers:

1. K. I. Mamis, New formulas for moments and functions of the multivariate normal distribution extending Stein's lemma and Isserlis theorem
2. G. A. Athanassoulis and K. I. Mamis, Extensions of the Novikov–Furutsu theorem, obtained by using Volterra functional calculus by
3. M. D. Donsker and J. L. Lions, Fréchet-Volterra variational equations, boundary value problems, and function space integrals
4. M. Scott, Applied Stochastic Processes in Science and Engineering, Lecture Notes, University of Waterloo, (2013).

Formula (45) of the first paper reads: $$\mathbb{E} \{ g [ X ] \, X_t \} = \mu ( t ) \, \mathbb{E} \{ g [ X ] \} + \int_0^T C ( t, s ) \, \mathbb{E} \left\{ \frac{\delta g[X]}{\delta X_s} \right\} \, ds,$$ where $$\delta / \delta X_s$$ is something called Volterra-Fréchet derivative, $$\mu$$ is the mean function and $$C(t, s)$$ is the covariance.

The third paper gives essentially the same formula (1.21): for a centered Gaussian process with covariance function $$\rho$$ holds $$\mathbb{E}_y^\rho \{ y ( \tau ) \, F [ y ] \} = \int_0^t \rho ( \tau, s ) \, \mathbb{E} \left\{ \frac{\delta F [ y ]}{\delta y ( s )} \right\} \, ds.$$ This paper also introduces the Volterra-Fréchet derivatives.

Finally, Donsker and Lions write that for the Brownian motion the formula $$\mathbb{E}_z^w \{ z ( \tau ) \, F [ z ] \} = \int_0^t \min ( \tau, s ) \, \mathbb{E} \left\{ \frac{\delta F}{\delta z ( s )} \right\} \, ds$$ goes back to the paper The first variation of an indefinite Wiener integral by Cameron.

I wonder if there is a modern standard framework for formulas like this? I still fail to see how they fit into the Malliavin calculus...