## Path Integrals and Feynman Diagrams for Classical Stochastic Processes

*15 Feb 2024 00:04*

This feels like a topic which should be obvious to me, but isn't, and so I want to wrap my head around it.

#### The parts I get

If we want to know the probability of a process with state space \( \mathcal{X} \) moving from state \( x \) at time \( 0 \) to state \( y \) at time \( t \), I need \( p(X(t) = y|X(0) = x) \). (Let's pretend everything has a density for now.) By the law of total probability I can insert as many intermediate times \( t_1, t_2, \ldots t_n \) as I like, and \[ p(X(t)=y|X(0)=x) = \int_{\mathcal{X}^n}{p(X(t_1) = u_1, X(t_2) = u_2, \ldots X(t_n) = u_n, X(t) = y|X(0) = x) du_{1:n}} \] which by the definition of conditional probability will be \[ p(X(t)=y|X(0)=x) = \int_{\mathcal{X}^n}{\left(\prod_{i=1}^n{p(X(t_i) = u_i|X(0)=x, X(t_1) = u_1, \ldots X(t_{i-1}) = u_{i-1}) } \right) du_{1:n}} \] with the understandings that \( t_{n+1} = t \), \( u_{n+1}=y \). If this is a Markov process, then earlier states become irrelevant when conditioning on later states, \[ p(X(t)=y|X(0)=x) = \int_{\mathcal{X}^n}{\left(\prod_{i=1}^n{p(X(t_i) = u_i| X(t_{i-1}) = u_{i-1})}\right) du_{1:n}} \] or \[ p(X(t)=y|X(0)=x) = \int_{\mathcal{X}^n}{\exp{\left(\sum_{i=1}^{n}{h(t_i, u_i| t_{i-1}, u_{i-1})} \right)} du_{1:n}} \] introducing the function \( h(t_i, u_i| t_{i-1}, u_{i-1}) \equiv \log{p(X(t_i) = u_i| X(t_{i-1}) = u_{i-1})} \). Assuming homogeneous transitions would then amount to assuming only the length of the interval \( t_i - t_{i-1} \) matters, so we could (overloading the notation a bit) write this as \( h(t_i, u_i| t_{i-1}, u_{i-1}) = h(u_i|u_{i-1}; t_{i} - t_{i-1}) \). A continuous, time-homogeneous Markov process will have a generator \( \mathbf{G} \), meaning that the transition operator over an interval \( \Delta t \) will be of the form \( e^{\Delta t \mathbf{G}} \). Taking \( \Delta t \) small, the transition operator will be \( \approx 1 + \Delta t \mathbf{G} \), and its log \( \approx \Delta t \mathbf{G} \), which would let us write \( h(u_i| u_{i-1}; \Delta t) \) in terms of the generator \( \mathbf{G} \) (at the cost of more algebra than I want to write down just now).

Passing non-rigorously to the limit, the sum inside the exponential will become an integral over time, and we should be able to write this over-all transition probability \( p(X(t)=y|X(0)=x) \) as a sum over all paths or histories, \[ p(X(t)=y|X(0)=x) = \int_{u: [0,1] \mapsto \mathcal{X}, u(0) = x, u(t)=y}{\exp{\left( \int_{s=0}^{t}{L(s, u(s), \dot{u}(s), \ldots) ds}\right)} du} \] where again \( L \) could be recovered from the generator if I was willing to do algebra. The presence of derivatives of the path in \( L \) comes from the fact that generators are (usually) differential operators.

(At this point a mathematical quibbler might well ask what \( du \) is, exactly, since we're now integrating over an infinite-dimensional space of continuous-time functions, and Lebesgue measure, for instance, doesn't properly extend to this setting. This is an important point I intend to ignore for the present.)

#### So what don't I understand?

Three big things stick out as especially irritating:- Where the cumulant generating function fits in to all this. (One problem here might be that while I can and do
*use*cumulants, I have no intuition about them at all.) - How to (in general) read off diagrammatic expansions from this.
- What to do for non-Markov processes. (Wio, below, suggests coming up with Markovian approximations.)

More broadly, I want to understand how much of this structure I learned as a
physicist really has anything to do with *physics*, and how much is just
a generality about stochastic processes.

- See also:
- Cumulants, and Cumulant Generating Functions
- Field Theory
- Interacting Particle Systems
- Large Deviations
- Nonequilibrium Statistical Mechanics
- Stochastic Differential Equations
- Stochastic Processes

- Recommended (very miscellaneous):
- Kirill Ilinski, Physics of Finance: Gauge Modelling in Non-equilibrium Pricing
- Richard D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem
- Eric Mjolsness, "Stochastic Process Semantics for Dynamical Grammar Syntax: An Overview", cs.AI/0511073
- Eric Smith, "Large-deviation principles, stochastic effective actions, path entropies, and the structure and meaning of thermodynamic descriptions", arxiv:1102.3938
- Horacio Wio, Path Integrals for Stochastic Processes

- To read:
- F. S. Abril and C. J. Quimbay, "Temporal fluctuation scaling in nonstationary time series using the path integral formalism", Physical Review E
**103**(2021): 042126 - Paul C. Bressloff, "Construction of stochastic hybrid path integrals using 'quantum-mechanical' operators", arxiv:2012.07770
- Yana A. Butko, René L. Schilling, Oleg G. Smolyanov, "Lagrangian and Hamiltonian Feynman formulae for some Feller semigroups and their perturbations", arxiv:1203.1199
- Joshua C. Chang, Van Savage, Tom Chou, "A path-integral approach to Bayesian inference for inverse problems using the semiclassical approximation", arxiv:1312.2974
- Vladimir Y. Chernyak, Mcihael Chertkov and Christopher Jarzynski, "Path-integral analysis of fluctuation theorems for general Langevin processes", cond-mat/0605471
- E. G. D. Cohen, "Properties of nonequilibrium steady states: a path integral approach", Journal of Statistical Mechanics (2008): P07014
- Leticia F. Cugliandolo, Vivien Lecomte, Frederic Van Wijland, "Building a path-integral calculus: a covariant discretization approach", Journal of Physics A: Mathematical and Theoretical
**52**(2019): 50LT01, arxiv:1806.09486 - Predrag Cvitanović
- Field Theory (1983)
- Quantum Field Theory: A Cyclist Tour (2005?)

- Ronald Dickman and Ronaldo Vidigal, "Path Integrals and Perturbation Theory for Stochastic Processes", cond-mat/0205321
- Cai Dieball, Aljaz Godec, "Feynman-Kac theory of time-integrated functionals: Ito versus functional calculus", arxiv:2206.04034
- S. H. Djah, H. Gottschalk, and H. Ouerdiane, "Feynman graphs for non-Gaussian measures", math-ph/0501030
- Masao Doi
- "Second quantization representation for classical many-particle system", Journal of Physics A
**9**(1976): 1465-1477 - "Stochastic theory of diffusion-controlled reaction",
Journal of Physics A
**9**(1976): 1479--1495

- "Second quantization representation for classical many-particle system", Journal of Physics A
- Josef Honerkamp, Stochastic Dynamical Systems
- Claude Itzykson and Jean-Michel Drouffe, Statistical Field Theory
- H. J. Kappen, "Path integrals and symmetry breaking for optimal control theory", Journal of Statistical Mechanics: Theory and Experiment (2005): P11011
- Richard Kleeman, "A path integral formalism for non-equilibrium Hamiltonian statistical systems", Journal of Statistical Physics
**158**(2015): 1271--1297, arxiv:1307.1102 - Hagen Kleinert, Path Integrals: In Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets
- Daniel C. Mattis and M. Lawrence Glasser, "The uses of quantum field theory in diffusion-limited reactions", Reviews of Modern Physics
**70**(1998): 979--1002 - L. Peliti, "Path integral approach to birth-death processes on a lattice", Journal de Physique
**46**(1985): 1469--1483 [Ungated copy] - Michael Polyak, "Feynman diagrams for pedestrians and mathematicians", math.GT/0406251
- Uwe C. Tauber, "Field Theory Approaches to Nonequilibrium Dynamics", cond-mat/0511743
- John J. Vastola, William R. Holmes, "Stochastic path integrals can be derived like quantum mechanical path integrals", arxiv:1909.1299
- Andreas Wipf, Statistical Approach to Quantum Field Theory