# Which functions are Wiener-integrable?

I'm looking for either a few precise mathematical statements about Wiener integrals, or a reference where I can find them.

## Background

The Wiener integral is an analytic tool to define certain "integrals" that one would like to evaluate in quantum and statistical mechanics. (Hrm, that's two different mechanics-es....) More precisely, one often wants to define/compute integrals over all paths in your configuration or phase space satisfying certain boundary conditions. For example, you might want to integrate over all paths in a manifold with fixed endpoints. It's conventional to write the integrand as a pure exponential $\exp(f)$. In statistical mechanics, the function $f$ in the exponent is generally real and decays to $-\infty$ (fast enough) in all directions. If the path space were finite-dimensional, you would expect such integrals to converge absolutely in the Riemann sense. In quantum mechanics, $f$ is usually pure-imaginary, so that $\exp(f)$ is a phase, and the integral should not be absolutely convergent, but in finite-dimensional integrals may be conditionally convergent in the Riemann sense. Typically, $f$ is a local function, so that $f(\gamma) = \int L(\gamma(t))dt$, where $\gamma$ is a path and $L(\gamma(t))$ depends only on the $\infty$-jet of $\gamma$ at $t$. In fact, typically $L(\gamma(t))$ depends only on the $1$-jet of $\gamma$, so that $f(\gamma)$ is defined on, for example, continuous piece-wise smooth paths $\gamma$.

For example, one might have a Riemannian manifold $\mathcal N$, and want to define: $$U(q_0,q_1) = \int\limits_{\substack{\gamma: [0,1] \to \mathcal N \\ {\rm s.t.}\, \gamma(0) = q_0,\, \gamma(1)=q_1}} \exp\left( - \hbar^{-1} \int_0^1 \frac12 \left| \frac{\partial \gamma}{\partial t}\right|^2dt \right)$$ where $\hbar$ is a positive real number (statistical mechanics) or non-zero pure imaginary (quantum mechanics). The "measure" of integration should depend on the canonical measure on $N$ coming from the Riemannian metric, and the Wiener measure makes it precise.

On $\mathcal N = \mathbb R^n$, I believe I know how to define the Wiener integral. The intuition is to approximate paths by piecewise linears. Thus, for each $m$, consider an integral of the form: $$I_m(f) = \prod_{j=1}^{m-1} \left( \int_{\gamma_j \in \mathbb R^n} d\gamma_j \right) \exp(f(\bar\gamma))$$ where $\bar\gamma$ is the piecewise-linear path that has corners only at $t = j/m$ for $j=0,\dots,m$, where the values are $\bar\gamma(j/m) = \gamma_j$ (and $\gamma_0 = q_0$, $\gamma_m = q_1$). Then the limit as $m\to \infty$ of this piecewise integral probably does not exist for a fixed integrand $f$, but there might be some number $a_m$ depending weakly on $f$ so that $\lim_{m\to \infty} I_m(f)/a_m$ exists and is finite. I think this is how the Wiener integral is defined on $\mathbb R^n$.

On a Riemannian manifold, the definition above does not make sense: there are generally many geodesics connecting a given pair of points. But a theorem of Whitehead says that any Riemannian manifold can be covered by "convex neighborhoods": a neighborhood is convex if any two points in it are connected by a unique geodesic that stays in the neighborhood. Then we could make the following definition. Pick a covering of $\mathcal N$ by convex neighborhoods, and try to implement the definition above, but declare that the integral is zero on tuples $\gamma_{\vec\jmath}$ for which $\gamma_j$ and $\gamma_{j+1}$ are not contained within the same convex neighborhood. This would be justified if "long, fast" paths are exponentially suppressed by $\exp(f)$. So hope that this truncated integral makes sense, and then hope that it does not depend on the choice of convex-neighborhood cover.

Of course, path integrals should also exists on manifolds with, say, indefinite "semi-"Riemannian metric. But then it's not totally clear to me that the justification in the previous paragraph is founded. Moreover, really the path integral should depend only on a choice of volume form on a manifold $\mathcal N$, not on a choice of metric. Then one would want to choose a metric compatible with the volume form (this can always be done, as I learned in this question), play the above game, and hope that the final answer is independent of the choice. A typical example: any symplectic manifold, e.g. a cotangent bundle, has a canonical volume form.

One other modification is also worth mentioning: above I was imagining imposing Dirichlet boundary conditions on the paths I wanted to integrate over, but of course you might want to impose other conditions, e.g. Neumann or some mix.

## Questions

Question 0: Is my rough definition of the Wiener integral essentially correct?

Question 1: On $\mathcal N = \mathbb R^n$, for functions $f$ of the form $f(\gamma) = -\hbar^{-1}\int_0^1 L(\gamma(t))dt$, for "Lagrangians" $L$ that depend only on the $1$-jet $(\dot\gamma(t),\gamma(t))$ of $\gamma$ at $t$, when does the Wiener integral make sense? I.e.: for which Lagrangians $L$ on $\mathbb R^n$, and for which non-zero complex numbers $\hbar$, is the Wiener integral defined?

Question 2: In general, what are some large classes of functions $f$ on the path space for which the Wiener integral is defined?

By googling, the best I've found are physics papers from the 70s and 80s that try to answer Question 1 in the affirmative for, e.g., $L$ a polynomial in $\dot\gamma,\gamma$, or $L$ quadratic in $\dot\gamma,\gamma$ plus bounded, or... Of course, most physics papers only treat $L$ of the form $\frac12 |\dot\gamma|^2 + V(\gamma)$.

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What is this Weiner integral? It gets only two hits on mathscinet, and at least one of the two is a misspelling of “Wiener”. –  Harald Hanche-Olsen Feb 20 '10 at 19:34
Yes, I think he means Wiener. –  Jonas Meyer Feb 20 '10 at 19:48
Well, that's embarrassing. But it's fixed now. –  Theo Johnson-Freyd Feb 20 '10 at 20:07
Have you looked here: books.google.com/books?id=Tti6Rr-Q_FUC (Johnson and Lapidus) –  Steve Huntsman Feb 20 '10 at 20:42
@Steve: No. I'll definitely look there when the library opens on Monday. Thanks! –  Theo Johnson-Freyd Feb 20 '10 at 22:32

1) I don't think there's a general theory. Trying a Lagrangian of the form $F = (\dot{\phi})^{1000}$ will not result in happiness. But: You can define Wiener measure using the standard kinetic term $\int \frac{1}{2}|\dot{\phi}|^2dt$ for any $\hbar$, and you can safely add a potential $V$ which is bounded below and integrable to the kinetic term.