Feynman–Kac Representation

• The Feynman–Kac representation is a framework that expresses solutions of partial differential equations using expectations over stochastic process trajectories and path integrals.
• It rigorously links analytic, geometric, and probabilistic methods by connecting PDE solutions with operator semigroups and stochastic processes, supporting both theory and numerical approaches.
• Modern extensions incorporate pseudo-differential operators, nonlinear control via BSDEs, and applications in infinite-dimensional and noncommutative settings to enhance computational techniques.

The Feynman-Kač (or Feynman–Kac) representation provides a foundational link between the solutions of partial differential equations (PDEs), stochastic processes, and operator semigroups via probabilistic and path integral formulations. Modern research extends this classical framework in several directions, encompassing pseudo-differential operators, nonlinear PDEs (notably Hamilton–Jacobi–BeLLMan equations), Feller and Lévy processes, vector bundle settings, noncommutative geometry, deformation quantization, and high-dimensional/infinite-dimensional analysis.

1. Classical Feynman–Kač Formula and Probabilistic Representations

The classical Feynman–Kač formula expresses the solution to a linear parabolic PDE (e.g., the heat equation) in terms of the expectation over stochastic process trajectories. For an initial value problem for the operator $\mathcal{L}$, such as

$$\partial_t u(t,x) = \mathcal{L} u(t,x) - V(x) u(t,x), \qquad u(0,x) = f(x)$$

with $\mathcal{L}$ as a second-order differential operator, the solution can be represented as

$$u(t,x) = \mathbb{E}^x \left[ \exp\left(-\int_0^t V(X_s) ds\right) f(X_t) \right]$$

where $X_t$ is the stochastic process generated by $\mathcal{L}$; in diffusive cases, $X_t$ is a Brownian motion with drift and volatility determined by $\mathcal{L}$. This underpins Monte Carlo methods for PDEs, exhibiting dimension-independent convergence properties (Pham, 2014).

The construction generalizes to pseudo-differential operators associated with Feller processes, where explicit transition densities are unavailable. In this context, the solution to the evolution equation or the associated semigroup $(T_t)$ may be represented as a path integral over the law of the Feller process, producing the archetypal Feynman–Kač representation (Butko et al., 2012).

2. Functional-Analytic Approaches: Feynman Formulae

To rigorously bridge infinite-dimensional functional integrals with computational and theoretical practice, the Feynman formulae express the operator semigroup as the strong limit of iterated finite-dimensional (multiple) integrals: $T_t = \lim_{n\to\infty} [F(t/n)]^n$ where $F(t)$ is a family of bounded operators, typically constructed via either Fourier (Hamiltonian) or transition kernel (Lagrangian) representations (Butko et al., 2012, Remizov, 2014).

Hamiltonian Feynman formula: In phase space, for a symbol $H(q,p)$,

$$F(t) \varphi(q) = (2\pi)^{-d/2} \int_{\mathbb{R}^d} e^{ip\cdot q} e^{-t H(q,p)} \hat{\varphi}(p) dp$$

which leads to the kernel representation as a finite-dimensional oscillatory integral, justified via Chernoff's theorem.

Lagrangian Feynman formula: In configuration space, for transition kernels $p_{t}(q, dy)$,

$$(T_t f)(q_0) = \lim_{n\to\infty} \int_{\mathbb{R}^{dn}} \prod_{k=1}^{n} p_{t/n}(q_{k-1}, dq_k) f(q_n)$$

which approximates the infinite-dimensional functional integral with a sequence of Markovian kernels.

These representations effectively replace infinite-dimensional objects with computable finite-dimensional integrals while preserving strong operator convergence. They enable numerical and analytic investigation of a broad class of semigroups, including those generated by non-local and pseudo-differential operators (Butko et al., 2012, Remizov, 2014).

3. Generalizations: Nonlinear PDEs and Stochastic Control

In nonlinear contexts, notably for fully nonlinear Hamilton–Jacobi–BeLLMan (HJB) equations and related integro-partial differential equations (IPDEs), the Feynman–Kač formula is generalized through backward stochastic differential equations (BSDEs) and their variants. For a nonlinear IPDE of HJB type, the probabilistic representation is given via the minimal solution of a constrained BSDE with jumps: $$Y_t = \xi + \int_t^T F(s, Y_s, Z_s, U_s) ds + K_T - K_t - \int_t^T Z_s dW_s - \int_t^T\int_E U_s(e)\tilde{\mu}(ds,de)$$ subject to $U_t(e) \le 0$ on the control set $A$. In a Markovian setting, the forward process includes both diffusion and jump (regime-switching) dynamics (Kharroubi et al., 2012): $$dX_s = b(X_s, I_s) ds + \sigma(X_s, I_s) dW_s + \int_\ell \beta(X_{s-}, I_{s-}, \ell) \tilde{\vartheta}(ds, d\ell), \qquad dI_s = \int_A (a - I_{s-}) \pi(ds, da)$$ The minimal BSDE solution yields the viscosity solution to the HJB IPDE, and the nonlinear Feynman–Kač representation is realized as $v(t,x) = Y_t$, independent of the auxiliary regime variable $a$ (Kharroubi et al., 2012, Pham, 2014).

A dual (weak control) formulation emerges by change of measure techniques (via the Doléans–Dade exponential), allowing the value function to be represented as an essential supremum over equivalent measures. This approach is particularly robust in degenerate, non-elliptic, or controlled-diffusion settings, where classical ellipticity is absent (Kharroubi et al., 2012, Bayraktar et al., 2016).

4. Path Integrals, Large Deviations, and Semiclassical Asymptotics

For generators given by nonlocal or integro-differential operators (e.g., Lévy generators), the Feynman–Kač representation applies to both configuration and momentum (Fourier) representations. The solution to evolution equations involving Lévy-type pseudo-differential operators is expressed as

$$u(t,p) = \mathbb{E}\left[ g(\xi_t) \exp\left(-\int_0^t U(\xi_s) ds\right) \mid \xi_0 = p\right]$$

where $\xi_t$ is a Lévy process and $U(p)$ is a nonnegative "killing rate" (Privault et al., 2013).

In the semiclassical regime ($\hbar \to 0$), large deviation theory provides precise asymptotic expansions for these expectations: $$u(t,p) \sim \exp\left(-\frac{1}{\hbar} \sup_{\phi \in \mathcal{A}} \left[-\int_0^t U(\phi(s) + p) ds - S(\phi)\right]\right)$$ where $S(\phi)$ is the rate function/action associated with the process. The minimizer corresponds to the classical path, and fluctuations around it are captured by quadratic corrections (prefactor analysis). This framework encompasses both diffusive and pure jump processes and rigorously connects path integral physics with probability theory (Privault et al., 2013).

5. Infinite-Dimensional, Geometric, and Noncommutative Extensions

The Feynman–Kač paradigm extends naturally to infinite-dimensional settings (e.g., parabolic PDEs on separable Hilbert spaces), with solutions constructed via Feynman formulae as strong limits of multiple Gaussian integrals

$$e^{tL}\varphi(x) = \lim_{n\to\infty} \left[S_{t/n}\right]^n \varphi(x)$$

where $S_{t}$ is an explicit Gaussian-integral operator involving the coefficients of $L$ (Remizov, 2014).

For vector bundles over Riemannian manifolds, the generalized Feynman–Kač formula incorporates parallel transport and Weitzenböck curvature terms, yielding stochastic representations for the heat kernel and generalized heat trace: $$P_t \phi(x) = \mathbb{E}^x [ \tau_{0,t} \phi(x(t)) \exp\{ \int_0^t V(x(u)) du \} ]$$ where $\tau_{0,t}$ denotes parallel translation, $V$ is an endomorphism-valued potential, and $x(t)$ is a Brownian motion or more general diffusion on the manifold. These representations allow for systematic expansion in geometric analysis and index theory (Ndumu, 2023).

For first-order (possibly non-selfadjoint) perturbations, the Feynman–Kač representation involves stochastic integrals of operator-valued quantities along the process, extending scalar approaches to handle operator semigroups relevant for noncommutative geometry. Explicit probabilistic representations for the traces involving differential operators furnish new formulas for invariants such as the equivariant Chern character, connecting to the Duistermaat–Heckman localization on loop spaces (Boldt et al., 2020).

6. Deformation Quantization, Path Integrals, and Spectral Asymptotics

In the deformation quantization framework, the Feynman–Kač representation is realized through the phase space star exponential: $$\operatorname{Exp}_*(H; t) = \sum_{n=0}^\infty \frac{(it)^n}{n!} H^{*n}$$ with Moyal/star-product powers. Integrating over phase space, after a Wick rotation $t \to -i\tau$, yields the Laplace sum over the spectrum: $$\frac{1}{2\pi\hbar} \int_{\mathbb{R}^2}\operatorname{Exp}_*(H; -i\tau)\,dx\,dp = \sum_n e^{-\tau E_n}$$ and the ground state energy is extracted as

$$E_0 = - \lim_{\tau\to\infty} \frac{1}{\tau} \ln \left[ \frac{1}{2\pi\hbar}\int_{\mathbb{R}^2}\operatorname{Exp}_*(H;-i\tau)\,dx\,dp \right]$$

This construction provides a deformation quantization analogue of the Feynman–Kač formula and a phase-space route for spectral analysis (Berra-Montiel et al., 5 Feb 2025).

7. Connections and Computational Perspectives

The Feynman–Kač representation and its generalizations serve as a bridge between analytic, geometric, and probabilistic methodologies, yielding:

• Operator semigroup representations via finite-dimensional integral approximations (especially for Feller, Lévy, and pseudo-differential generators), with rigorous convergence guaranteed by advanced functional-analytic tools (e.g., Chernoff's theorem for operator approximants) (Butko et al., 2012, Remizov, 2014).
• Nonlinear stochastic control representations for fully nonlinear PDEs via forward-backward SDEs with constraints, leading to randomized numerical schemes compatible with high-dimensional settings and robust to degeneracies (Kharroubi et al., 2012, Pham, 2014, Bayraktar et al., 2016).
• Explicit analytic and probabilistic formulas for eigenvalue problems, heat kernel expansions, and geometric invariants, which penetrate index theory, noncommutative geometry, and quantum field theory (Boldt et al., 2020, Ndumu, 2023).
• Numerical methods based on particle systems and interacting diffusions for high-dimensional and non-conservative (generalized Fokker–Planck) PDEs, leveraging the McKean Feynman–Kač paradigm (Izydorczyk et al., 2019).
• Algebraic dualities in Feynman integral representations and connections to momentum-space and Baikov parametrizations, which clarify the structural symmetries of scattering amplitudes and quantum field theoretical calculations (Chen, 2023).

The Feynman–Kač representation thus provides a versatile and rigorous conceptual tool, supporting both theoretical advances and simulation methodologies in mathematical physics, stochastic analysis, partial differential equations, geometric analysis, and quantum field theory.