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The Feynman-Kac formula says that $$ \exp(-t(-\Delta+V(X)))(x,y) = \mathbb{E}_{\gamma(0)=x,\gamma(t)=y}\left[\exp(-\int_0^t V\circ\gamma)\right] $$ where $\Delta$ is the Laplacian on $L^2(\mathbb{R}^n)$, $X$ is the position operator on $L^2(\mathbb{R}^n)$, $V:\mathbb{R}^n\to\mathbb{R}$ is some potential function, $x,y\in\mathbb{R}^n$, and $\mathbb{P}_{\gamma(0)=x,\gamma(t)=y}$ is the conditional Wiener measure for a Brownian path $\gamma$.

Question 1

What is the discrete-time analog of this formula? I mean what are the relevant notions on both the LHS and RHS?

Question 2

What is the discrete-space analog of this formula (at continuous time)? I mean, what is the analogous LHS if we change the RHS to make $\gamma$ take values only on $\mathbb{Z}^n$ instead of $\mathbb{R}^n$? I think this might be answered in the body of this question.

Question 3

We know that the Gaussian free field (GFF) coincides with the Wiener measure if the domain space happens to be one dimensional. However, if the domain is more than one dimensional, what is the analog of the Feynman-Kac formula? What I mean is the following. Let $\varphi:D\to\mathbb{R}^n$ where $D$ is a compact subset of $\mathbb{R}^m$. Consider the probability measure on random fields $\varphi$ proportional to $$ \exp(-\int_D\varphi\cdot(-\Delta\varphi))\,. $$ Here, the inner product is meant in $\mathbb{R}^n$, and the Laplacian is the Dirichlet Laplacian on $D$. This defines the GFF measure on fields $D\to\mathbb{R}^n$. One way to understand it is via a spectral eigenfunction decomposition (in Dirichlet eigenfunctions of the Laplacian on $D$) of the field $\varphi$. If $D=[0,t]$ this yields (pinned) Brownian motion and we have a Feynman-Kac formula. What would, then, be the analogous LHS of a putative Feynman-Kac formula for the GFF? What I mean is:

$$ \boxed{?} = \frac{\int_{\varphi:D\to\mathbb{R}^n}\exp(-\int_D\varphi\cdot(-\Delta\varphi)-\int_D V\circ\varphi)\mathrm{d}\varphi}{\int_{\varphi:D\to\mathbb{R}^n}\exp(-\int_D\varphi\cdot(-\Delta\varphi))\mathrm{d}\varphi}\,. $$

Here $V:\mathbb{R}^n\to\mathbb{R}$ would be some potential function.

What is the analogous heat equation PDE? Can this work if we replace $\mathbb{R}^n$ with $\mathbb{Z}^n$ (related to Question 2)

I suppose when $D=[0,t]\times\tilde{D}$ then one may write a heat-kernel for a Schroedinger operator on the Hilbert space $L^2(\tilde{D})$? Similar to the Euclidean path-integral prescription of QFT where we have a field that isn't just a point particle. And then the field obeys something like a heat-Klein-Gordon equation? But what if $D$ is general? And is this rigorous as the usual Feynman-Kac formula is?

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Question 1

The keyword to look up is Chernoff's Theorem. Essentially, you can take an approximation $\mu_t$ of the Wiener measure for $\Delta + V$ and consider for a partition $t = t_1 + \dots + t_N$ of the interval $[0,t]$ the measure $\mu_{t_1}*\cdots*\mu_{t_N}$ which is obtained by integrating the restrictions of a path $\gamma$ to $[t_1+\dots+t_{i-1},t_1+\dots+t_{i-1} + t_{i}]$ against $\mu_{t_i}$ and taking the product. As long as $\mu_{t}$ approximates the Wiener measure to first order, in the sense that $\mathbb E_{\mu_t}(f(\gamma(0))g(\gamma(t))) = \langle f - t(\Delta + V)f,g\rangle + O(t^2)$, the limit of $\mu_{\{t_i\}}$ where the size of the partition goes to zero exists and is the Wiener measure. Convenient choices are:

  • $\mu_t(\gamma) = e^{-tV(\gamma(0))}\mu_t^0$, where $\mu_t^0$ is the Wiener measure with $V = 0$; the resulting identity is the Feynman-Kac formula
  • $\mu_t(\gamma)$ is concentrated on geodesics, i.e. straight lines (for a general manifold, one can take only those with length smaller than the injectivity radius), and $\mu_t(\gamma) \sim t^{-n/2}e^{-d(\gamma(0),\gamma(t))^2/t}\mathrm d\gamma(0)\mathrm d\gamma(t)$ is the flat heat kernel. This gives Brownian motion on the target Riemannian manifold as a limit of random walks, with idd Gaussian steps in the flat case.
  • One can combine both approaches to obtain finite-dimensional approximations to Brownian motion on a Riemannian manifold with a potential.

Question 2

The question that you reference seems like a reasonable summary. Essentially, you can discretize the target by choosing a finite set of points in the target, whose pairwise distances generate a weighted graph. Random walks on this graph converge to Brownian motion when the finite set approximates the manifold better and better; for $\mathbb R^n$, this is just Donsker's theorem which says that a rescaled limit of random walks converges to Brownian motion (together with Chernoff's theorem to relate random walks on the graph to the diffusion process on it if you only want to approximate the target). Compare this question for some references which make this statement precise.

Question 3

After these successes, you now want to apply these techniques to the Gaussian free field, as well as the interacting version that you get by adding a potential term. This runs into many problems:

  • You can discretize the source and write down the formal functional integral (which concentrates on classical solutions, i.e. harmonic maps), but Chernoff's theorem does not hold anymore and the limit as the mesh size of the discretization goes to zero does not exist. Physicists tell us that one should take a "renormalized" limit, where you subtract some singular terms at each stage to get a well-defined limit. However, even if this works the result will depend on the chosen discretization and regularization; for instance, the result might not be scale-invariant even if the integral expression looks like it was (physicists call this the $\beta$ function).
  • For the Feynman-Kac formula, you want to exhibit the "field integral measure" of the interacting field theory defined by the potential $V$ as absolutely continuous with respect to the measure of the Gaussian free field. In your proposed formula, the Radon-Nikodym derivative is proportional to $e^{-\int_D V(\phi(x))\mathrm dx}$. However, this formula does not make sense for $\dim D > 1$, since $\phi$ is a random distribution which is continuous with probability zero, so that the expression $V(\phi)$ does not make sense for interesting interaction terms $V$ since distributions can't be multiplied in general. If you ignore these issues and just formally calculate with the Feynman-Kac formula, you can derive Feynman's diagram expansion with all of its famous singularities. A more rigourous approach is to replace $V(\phi)$ by an effective action, which is a regularization which makes sense for $\phi$ a distribution. One then has to show that the result is independent of all choices. The perturbative expansions are understood fairly well mathematically, see for instance the book Factorization Algebras in Quantum Field Theory by Costello and Gwilliam. In the non-perturbative realm there are also plenty of results, see for instance this recent paper by Barashlov and Gubinelli which builds the functional measure for $\Phi^4$-theory in three dimensions and show that the result is not absolutely continuous with respect to the non-interacting theory.
  • Formally, the field integral over maps from $\widetilde D\times [0,t]$ to $M$ should be the Wiener measure on the path space of the infinite-dimensional manifold $\operatorname{Map}(\widetilde D,M)$. Of course this does not actually exist for all of the above reasons. For a general $D$, the field integral should define an element of a Hilbert space associated to the boundary, and the derivative of this element with respect to a deformation of the boundary is a quantization of the observable obtained by applying Noether's theorem to translation invariance (in $d$ dimensions, Noether's theorem produces a current/$(d-1)$-form from a symmetry, which needs to be integrated over a closed $(d-1)$-manifold to obtain an observable). Of course you again have to regularize to get a rigourous statement, which might not be possible and depend on choices.
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