Path integral formalization using measure-based integration and equivalence with limit-based definitions

Question

This question is coming from a computational physicist who is very comfortable with numerical and computational math, but much less comfortable with distributions, measures, and Lebesgue integration. The context of this question is that I'm trying to formalize and generalize the Hubbard-Stratonovich transformation [https://doi.org/10.1103/PhysRevLett.3.77], which inevitably involves addressing formal aspects of path integrals. However, I'll ask my question using a simpler representative formula to avoid unnecessary complications. I've certainly tried to answer this question on my own already, but I've mostly saturated the level of understanding that I can attain on my own without investing a substantially larger amount of time. A Math StackExchange post feels like the next logical step, and hopefully I am now capable of asking a well-posed question.

The representative formula is

$$\lim_{n \rightarrow \infty} \int_{\mathbb{R}^n} f(x_1 \theta_{[0,1/n]} + x_2 \theta_{(1/n,2/n]} + \cdots + x_n \theta_{(1-1/n,1]}) \exp[ - (x_1^2 + \cdots + x_n^2)/n] dx_1 \cdots dx_n$$

where $f$ is a map from real-valued functions on $[0,1]$ to real values, and $\theta_{A}$ are step/indicator functions for $A \subseteq [0,1]$. While I don't yet know what precise properties to assign to $f$, I'm assuming that I can evaluate this integral for finite values of $n$ and that the $n \rightarrow \infty$ limit is well-defined. As a practical matter, this is likely how I would evaluate such expressions. However, I'd like to know if this is formally equivalent to a Lebesgue integral that doesn't involve an explicit limit,

$$ \int_{\mathcal{F}} f(x) d\mathcal{\gamma}(x) $$

for some measure $\gamma$ on some function space $\mathcal{F}$ with the domain $[0,1]$, and if there are any tools/formulas for evaluating such integrals without the use of limiting expressions. Presumably, most tools/formulas would depend on the structure of $f$. A specific example of what I expect to be a simple, well-behaved $f$ here is

$$ f(x) = \left| \int_{[0,1]} x(t) g(t) dt \right|^2 $$

where $g$ is an absolutely integrable function on $[0,1]$, but I certainly don't want to be restricted to this form.

First, can this limit be described as the Lebesgue integration of $f$ with respect to an infinite-dimensional Gaussian measure defined using an abstract Wiener space [https://en.wikipedia.org/wiki/Abstract_Wiener_space]? I am still quite far from understanding this theory, and the most intelligible authoritative source for me so far has been Stroock's recent book (Gaussian Measures in Finite and Infinite Dimensions). A concerning detail that comes up in this book (in the section "Gaussian measures on a Hilbert space") and other sources are trace-class/nuclear requirements on the covariance operator of the Gaussian measure in the context of the parent Hilbert space. Does the covariance operator need to have a finite trace to be defined as a measure on the Hilbert space without using the abstract Wiener space construction, or is that also a requirement to be defined as a measure on the encompassing Banach space?

Second, can this limit be described using "white noise analysis" [https://en.wikipedia.org/wiki/White_noise_analysis] and its white noise probability measure (or some variant)? Also, is "white noise analysis" an application of the abstract Wiener space or a distinct theory? This measure is defined on the space of tempered distributions, and it isn't clear to me that you'd get an integral with the same value when integrating over this space. Also, it is a space of distributions rather than functions, so I'm not sure that my problem is well-posed here without assuming that $f$ extends to the space of distributions somehow. On the other hand, I'm trying to integrate over functions on $[0,1]$ rather than $\mathbb{R}$, which might ameliorate these problems somehow.

I think the deeper question here is whether or not Gaussian white noise has enough structure to define a measure on a usefully large function space with a compact domain, or if the functions being integrated have to be overly well-behaved (i.e. a failure of the typical separation into measures and measurable functions of Lebesgue integration).

Addendum in response to Abdelmalek Abdesselam's answer: My example of a well-behaved $f$ in the original post was poorly chosen in hindsight and likely isn't well-behaved at all. After considering examples more carefully, a better example would be:

$$f(x) = \left\{ \begin{array}{cc} 1, & 0 \le x(t) \le 1 \ \forall t \in [0,1] \\ 0, & \mathrm{otherwise} \end{array} \right.$$

which should be $\exp(-1/3)$ in the $n \rightarrow \infty$ limit. The point of an example is that this limit can be well-behaved with the current choice of normalization, which will not lead to a probability measure. Part of why I'm asking this question is to understand the possibility of a measure on a function space that doesn't have finite total variation. Otherwise, I have to rely on special structure in $f$ to define a measure, presumably with finite total variation, and the exponential terms would then be the function being integrated against a measure defined by $f$.

Answer 1

I don't know what was the issue with functionals being linear or not. A lot of mathematicians are totally fine with nonlinear functionals.

As to the question, one first needs to make a small correction. A simple functional for which one would like the construction to work is $f(x)=1$, the constant functional equal to one. Now, as it is written, the integral would then be equal to $(\pi n)^{\frac{n}{2}}$ which does not converge as $n\rightarrow \infty$. This is a minor issue which can be fixed by correctly normalizing the probability measure one is starting from.

Let $\gamma_n$ be the probability measure on $\mathbb{R}^n$ given by $$ (\pi n)^{-\frac{n}{2}} \exp\left[-\frac{1}{n}(x_1^2+\cdots+x_n^2)\right]\ dx_1\cdots dx_n\ . $$ Leaving the choice of function space $\mathcal{F}$ blank for now, we define an "extension" map ${\rm ex}_n:\mathbb{R}^n\rightarrow\mathcal{F}$, which sends the tuple $(x_1,\ldots,x_n)$ to the function $$ x=x_1\theta_{\left[0,\frac{1}{n}\right]}+x_2\theta_{\left(\frac{1}{n},\frac{2}{n}\right]}+\cdots x_n\theta_{\left(\frac{n-1}{n},1\right]}\ . $$ The integral, with the proper normalization, then becomes $$ \int_{\mathcal{F}}f(x)\ d\mu_n(x) $$ where $\mu_n:=({\rm ex}_n)_{\ast}\gamma_n $ is the push-forward or direct image measure. So the problem becomes that of the weak convergence of the sequence of probability measures $\mu_n$.

Now let's make this more precise. To have a lot of room (making it hard for probability measure have mass escape to infinity), we will pick a rather big function space, namely $\mathcal{F}=\mathscr{S}'(\mathbb{R})$, the space of temperate Schwartz distributions on $\mathbb{R}$. We will redefine the extension map as follows. The Schwartz distribution $T={\rm ex}_n(x_1,\ldots,x_n)$, is the one which sends a Schwartz function $g\in\mathscr{S}(\mathbb{R})$ to $$ \langle T, g\rangle:=\int_{[0,1]}\left\{ x_1\theta_{\left[ 0,\frac{1}{n}\right]}(t)+x_2\theta_{\left(\frac{1}{n},\frac{2}{n}\right]}(t)+\cdots x_n\theta_{\left(\frac{n-1}{n},1\right]}(t) \right\}g(t)\ dt $$ i.e., to $$ \langle T, g\rangle=\sum_{j=1}^n x_j \int_{\frac{j-1}{n}}^{\frac{j}{n}} g(t)\ dt\ . $$ Weak convergence of probability measures on a space, requires a topology, so we will put on $\mathcal{F}=\mathscr{S}'(\mathbb{R})$ the canonical topology which is the strong dual topology, and the $\sigma$-algebra we will use is the Borel $\sigma$-algebra for that topology.

Consider the characteristic function $\Phi_n$ of the probability measure $\mu_n$, namely, the function from $\mathscr{S}(\mathbb{R})$ to $\mathbb{C}$ given by $$ \Phi_n(g)=\int_{\mathscr{S}'(\mathbb{R})} e^{i\langle T,g\rangle}d\mu_n(T)\ . $$ We have $$ \Phi_n(g)=\int_{\mathbb{R}^n} e^{i \langle {\rm ex}_n(x_1,\ldots,x_n),g\rangle} d\gamma_n(x_1,\ldots,x_n) $$ $$ =(\pi n)^{-\frac{n}{2}} \int_{\mathbb{R}^n} \exp\left(i\sum_{j=1}^n x_j \int_{\frac{j-1}{n}}^{\frac{j}{n}} g(t)\ dt\right) \exp\left[-\frac{1}{n}(x_1^2+\cdots+x_n^2)\right]\ dx_1\cdots dx_n\ . $$ $$ =\exp\left[-\frac{n}{4}\sum_{j=1}^{n}\left(\int_{\frac{j-1}{n}}^{\frac{j}{n}} g(t)\ dt\right)^2 \right] $$ which converges when $n\rightarrow\infty$ to $$ \Phi(g):=\exp\left[-\frac{1}{4}\int_0^1 g(t)^2\ dt\right] $$ which is the characteristic function of a multiple of white noise supported on the interval $[0,1]$. The above proves that the measures $\mu_n$ converge weakly to the unique (Gaussian) probability measure $\mu$ on $\mathscr{S}'(\mathbb{R})$ with characteristic function given by $\Phi$.

Answer 2

Relevant to your first question, check out reproducing kernel Hilbert spaces (RKHS). If an RKHS is separable, given a mean function, you can always construct a Guassian measure with the given mean function and covariance. See this question for reference construction of Gaussian measure with a given mean function and covariance kernel

After that, you might be interested in section 3.5 of the book "Mathematical Feynman Path Integrals and their Applications" by Mazzucchi. The idea is that in the usual construction of the path integral, there is no complex measure since these finite dimensional steps on the way to the limit are not bounded in total variation.

Note that I am not familiar with these tools, so it may be the case that you could consider other covariance kernels (other than the one corresponding to the Wiener process) or considering compact spaces instead of $\mathbb{R}^n$ to get a limit which can be represented by a complex measure and locally looks like the usual Wiener measure when you go from complex time to real time, but I am just spitballing.