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From the Chertkov wiki — draft for Misha

Harmonic Path Integral Diffusion

Harmonic Path Integral Diffusion (H-PID) is an exactly solvable generative sampling algorithm that transports a Dirac initial condition to a prescribed terminal probability distribution along a stochastic bridge, and computes the optimal drift analytically through the closed-form Green functions of a quadratic potential Behjoo & Chertkov (2025). H-PID is the quadratic-potential specialisation of Path Integral Diffusion (PID), a linearly-solvable stochastic optimal control formulation of the Schrödinger bridge problem Kappen (2005); Chernyak, Chertkov, Bierkens & Kappen (2014). Because the Green functions are Gaussian, the optimal drift is available without training a neural network, in contrast with score-based diffusion models Song et al. (2021).

Background: Path Integral Diffusion

Given a target density \(p^{\mathrm{tar}}\) on \(\mathbb{R}^d\), Path Integral Diffusion seeks a controlled Itô diffusion

\[ dx_t = u_t(x)\,dt + dW_t,\qquad x_0 = 0,\qquad \mathrm{Law}(x_1) = p^{\mathrm{tar}}, \]

that minimises the stochastic-optimal-transport cost

\[ C[u] = \mathbb{E}\!\left[\int_0^1 \left(\tfrac{1}{2}\|u_t\|^2 + V_t(x_t)\right) dt\right] \]

subject to the hard terminal-law constraint Behjoo & Chertkov (2025). Under a quadratic control cost and a state potential \(V_t(x)\), the Hamilton–Jacobi–Bellman (HJB) equation for the value function linearises via the Hopf–Cole substitution \(J = -\log\psi\) into an imaginary-time Schrödinger equation for \(\psi\) Kappen (2005). The optimal drift is then

\[ u^*_t(x) = \nabla_x \log \psi_t(x), \]

and admits a Feynman–Kac path-integral representation, from which PID inherits its name Kappen (2005); Behjoo & Chertkov (2025). The gauge-invariant extension of this machinery was developed in Chernyak, Chertkov, Bierkens & Kappen (2014), and its discrete-state counterpart is the linearly-solvable Markov decision problem of Todorov (2007).

The harmonic potential

H-PID specialises PID to a quadratic, positive-definite potential

\[ V_t(x) = \tfrac{\beta}{2}\,\|x\|^2,\qquad \beta > 0, \]

so that the linearised operator is the Hamiltonian of the quantum harmonic oscillator Behjoo & Chertkov (2025). The backward and forward Green functions \(G_-(t,x;1,y)\) and \(G_+(t,x;0,0)\) of this operator are Gaussian kernels with hyperbolic time-dependence, obtained through the Mehler formula. The drift takes the form

\[ u^*_t(x) = \nabla_x \log \int dy\; p^{\mathrm{tar}}(y)\,G_-(t,x;1,y)\big/G_+(1,y;0,0), \]

which reduces to an affine combination of the current state \(x\) and a re-weighted conditional mean of the terminal state Behjoo & Chertkov (2025). All time-dependent coefficients are explicit functions of \(\sinh\), \(\cosh\), and \(\coth\) of \(\sqrt{\beta}\,(1-t)\), which is why the harmonic case is sometimes described as the "solvable hydrogen atom" of generative diffusion Behjoo & Chertkov (2025).

Exact solvability

Behjoo & Chertkov (2025) organise integrability of the PID family into three nested levels:

  1. General bounded potentials. The Hopf–Cole transform yields a formal Green-function representation of \(u^*_t\) that can in principle be evaluated by numerical path integration.
  2. Quadratic positive-definite potentials. The Green functions are Gaussian, and \(u^*_t\) becomes an affine map in \(x\) with coefficients given by hyperbolic functions of \(t\). This is the H-PID regime.
  3. Uniform quadratic potential with zero forcing and zero gauge. The drift reduces to convolutions of the target with Gaussian kernels, giving the simplest closed-form sampler.

Within levels 2 and 3 the drift evaluation reduces to Gaussian integration against the target, requiring no neural network: all operations execute as sequential matrix-vector products and conditional branching, and in particular run on a standard CPU Behjoo & Chertkov (2025). Behjoo & Chertkov (2025) further identify a time-independent Hessian for the value function, enabling Gaussian importance sampling from targets known only up to normalisation.

Relation to the Schrödinger bridge

H-PID realises a specific analytically tractable instance of the Schrödinger bridge problem — the minimum-relative-entropy coupling of path laws between a fixed initial distribution (here a Dirac at the origin) and a prescribed terminal distribution. In the Schrödinger–Doob formulation Kappen (2005), the bridge drift is the gradient of a logarithmic Doob \(h\)-transform of a reference diffusion; in PID this reference is enriched by a state potential \(V_t\) and, in the gauge-invariant extension, a vector potential \(A_t\) Chernyak, Chertkov, Bierkens & Kappen (2014). Whereas the general Schrödinger bridge requires iterative solution (for example via Sinkhorn-like schemes in neural implementations), the harmonic potential collapses the fixed-point iteration into a single Gaussian integral, which is what makes H-PID exactly solvable Behjoo & Chertkov (2025).

Relative to score-based generative modelling Song et al. (2021), H-PID's analytical drift plays the role of the learned score \(\nabla_x \log p_t\), but is computed in closed form rather than trained on samples Behjoo & Chertkov (2025).

Extensions

Two subsequent extensions preserve the linear solvability of the harmonic case while relaxing its rigidity.

Both extensions remain within the integrability hierarchy of Behjoo & Chertkov (2025).

See also

References