1.1 Background

Newton’s discovery of differential equations and calculus was crucial in developing classical mechanics because it allowed for the mathematical description of the motion of objects. This discovery took a groundbreaking step in unifying mathematics with physics, enabling the prediction of planetary orbits, the motion of objects under various forces, and much more, and marked the beginning of a new era in mathematics and science, laying the cornerstone for over three centuries of advancements.

Newton understood the immediate impact of his discoveries and their potential to transform the understanding of the natural world. To establish and protect his intellectual property rights at the same time, he concealed his discovery in the fundamental anagram of calculus, which he included in his 1676 letter to Oldenburg. This anagram contained a Latin statement describing the method of fluxions (his term for calculus) when decoded. The need for an anagram reflected that Newton was competitive and cautious in equal measure by balancing the desire for recognition with the fear of disclosure. The number of occurrences of each Latin character in Newton’s sentence agrees with his anagram, thus proving that the actual sentence was written in 1676.Footnote ² The original letter is shown in Figure 1.

Figure 1 Newton’s letter to Oldenburg, 1676.

Reproduced by kind permission of the Syndics of Cambridge University Library.

The fact that differential equations are instrumental in mathematics and physics alike was firmly established in the late seventeenth century. However, methods for solving these equations remained ad hoc for more than a century until the work by Lagrange, Laplace, Fourier, and many other mathematicians and physicists. In particular, the Fourier transform stands out as the most potent tool in an applied mathematician’s toolkit, enabling the solving of linear partial differential equations (PDEs) and partial pseudo-differential equations (PPDEs) with spatially constant coefficients; it is also invaluable for analyzing time series and tackling other critical tasks (Reference FourierFourier (1822); Reference Morse and FeshbachMorse & Feschbach (1953)).

At the heart of the $n$-dimensional Fourier method are wave functions, expressed as follows:

$ℱ\left(t,x,k\right)=a\left(t\right)\exp \left(ik\cdot x\right)\text{},$(1.1)

where $\mathbf{x}$ and $\mathbf{k}$ are $n$-dimensional vectors, $\cdot$ denotes the scalar product, $\mathsf{a}\left(t\right)$ is the amplitude, and $\mathbf{k}\cdot \mathbf{x}$ is the phase. Depending on the particular problem at hand, the amplitude $\mathsf{a}\left(t\right)$ can be a scalar or a vector, hence the notation. Substituting $\mathcal{F}$ into a PDE with spatially constant coefficients, one reduces the problem of interest to a system of ordinary differential equations (ODEs) or a single ODE when $\mathsf{a}\left(t\right)$ is scalar. Of course, this system parametrically depends on $\mathbf{k}.$

This Element studies PDEs and PPDEs with coefficients linearly dependent on $\mathbf{x},$ which are called affine. Hence, one must use a more general approach and consider wave functions with time-dependent wave vectors: Reference KelvinKelvin (1887) and Reference OrrOrr (1907) were the first to use such waves to analyze the stability of the steady motions of an incompressible fluid.

$\begin{array}{r}\mathcal{K}\left(t,\mathbf{x}\mathbf{,}\beta \left(t\right)\right)=\mathsf{a}\left(t\right)exp\left(i\mathit{\beta }\left(t\right)\cdot \mathbf{x}\right).\end{array}$(1.2)

Affine problems are not artificial constructs. They appear organically in several situations, for example, when the linear description of the underlying physical mechanism is either exact or provides an excellent approximation to reality or when the evolution in the phase space is studied; see Section 3.

Subsequently and independently, affine PDEs and the associated wave functions were used by many researchers in various areas, including the theory of stochastic processes, physics, biology, and mathematical finance, to mention a few. The Ornstein–Uhlenbeck (OU) and Feller processes are the simplest but extremely important examples of affine processes; see Reference Uhlenbeck and OrnsteinUhlenbeck and Ornstein (1930), Reference ChandrasekharChandresekhar (1943), and Reference FellerFeller (1951, Reference Feller1952). For financial applications of affine processes see Reference Duffie and KanDuffie and Kan (1996), Reference Duffie, Pan and SingletonDuffie et al. (2000), Reference Dai and SingletonDai and Singleton (2000), Reference LiptonLipton (2001), Reference Duffie, Filipovic and SchachermayerDuffie et al. (2003), Reference SeppSepp (2007), Reference Lipton and SeppLipton and Sepp (2008), and Reference FilipovicFilipovic (2009), among others.

This Element uses Kelvin waves of the form (1.2) to study transition probability density functions (t.p.d.fs) for affine stochastic processes. These processes can be either degenerate, namely, have more independent components than the sources of uncertainty, or nondegenerate, when every component has its source of uncertainty. Recall that the t.p.d.f. for a stochastic process describes the likelihood of a system transitioning from one state to another over a specified period. Knowing the iterated t.p.d.f. is fundamental for understanding the dynamics and behavior of stochastic processes over time and is tantamount to knowing the process itself.

In this Element, Kelvin waves are also used to solve several essential and intricate problems occurring in financial applications. These include pricing options with stochastic volatility, path-dependent options, and Asian options with geometric averaging, among many others.

The main objective is to link various financial engineering topics with their counterparts in hydrodynamics and molecular physics and showcase the interdisciplinary nature of quantitative finance and economic modeling. Finding such connections allows us to understand better how to model, price, and risk-manage various financial instruments, derive several new results, and provide additional intuition regarding their salient features. This Element continues previous efforts in this direction; see Reference Lipton and SeppLipton and Sepp (2008) and Reference LiptonLipton (2018), chapter 12.

There are several approaches one can use to solve affine equations efficiently. For instance, Lie symmetries are a powerful tool for studying certain classes of affine equations. Numerous authors describe general techniques based on Lie symmetries; see, for example, Reference OvsiannikovOvsiannikov (1982), Reference IbragimovIbragimov (1985), Reference OlverOlver (1986), and Reference Bluman and KumeiBluman and Kumei (1989), while their specific applications to affine equations are covered by Reference BerestBerest (1993), Reference AksenovAksenov (1995), Reference Craddock and PlatenCraddock and Platen (2004), Reference CraddockCraddock (2012), and Reference Kovalenko, Stogniy and TertychnyiKovalenko et al. (2014), among many others. However, Lie symmetry techniques are exceedingly cumbersome and might be challenging to use in practice, especially when complicated affine equations are considered.

Laplace transform of spatial variables can be used in some cases, for instance, for Feller processes; see, for example, Reference FellerFeller (1951, Reference Feller1952). However, they are hard to use for solving generic affine equations.

Reductions of a given equation to a simpler, solvable form is another powerful method that can be successfully used in many instances; see, for example, Reference ChandrasekharChandresekhar (1943), Reference Carr, Lipton and MadanCarr et al. (2002), Reference Lipton, Gal and LasisLipton et al. (2014), and Reference LiptonLipton (2018), chapter 9. Although the reduction method is quite powerful, experience suggests it is often hard to use in practice.

Finally, the affine ansatz based on Kelvin waves provides yet another approach, which is the focus of the present Element; see also Reference Duffie and KanDuffie and Kan (1996), Reference Dai and SingletonDai and Singleton (2000), Reference Duffie, Filipovic and SchachermayerDuffie et al. (2003), Reference Lipton and SeppLipton and Sepp (2008), Reference FilipovicFilipovic (2009), and Reference LiptonLipton (2018), chapter 12. Undoubtedly, the affine framework, also known as the affine ansatz, is the most potent among the abovementioned techniques due to its comprehensive nature, versatility, and (relative) ease of use, even in complex situations. In practice, applications of Kelvin waves consist of three steps:

• Effectively separating variables for the evolution problems with pseudo-differential generators linearly dependent on spatial coordinates;

• Solving ODEs parametrized by time-dependent wave vectors; see (1.2);

• Aggregating their solutions together to get the solution to the original problem.

However, despite being a ruthlessly efficient tool, Kelvin waves have limitations – using them to solve evolution problems supplied with external boundary conditions is challenging. This exciting topic is being actively researched now; it will be discussed elsewhere in due course.

1.2 Main Results

This Element develops a coherent, unified mathematical framework using Kelvin waves as a powerful and versatile tool for studying t.p.d.fs in the context of generic affine processes. It discovers previously hidden connections among large classes of apparently unrelated problems from hydrodynamics, molecular physics, and financial engineering. All these problems require solving affine (pseudo-) differential equations, namely, equations with coefficients, which linearly depend on spatial variables. The Element discusses some classical results and derives several original ones related to:

• small wave-like perturbations of linear flows of ideal and viscous fluids described by Euler and Navier–Stokes equations, respectively;

• motions of free and harmonically bound particles under the impact of random external white-noise forces described by the Klein–Kramers equations and the hypoelliptic Kolmogorov equation, which play an essential role in statistical physics;

• Gaussian and non-Gaussian affine processes, such as the Ornstein–Uhlenbeck and Feller processes, which are the archetypal mean-reverting processes, and their generalizations;

• dynamics of financial markets, particularly derivative products.

To solve some of the more complicated problems, one must augment primary processes by introducing subordinate processes for auxiliary variables, such as integrals over the original stochastic variable, and develop a uniform mathematical formalism to construct t.p.d.fs for the abovementioned processes.

Quite unexpectedly, the analysis identifies and rectifies an error in the original solution of the Kolmogorov equation. The rectified solution is dimensionally correct, properly scales when the process parameters change, and agrees with numerical results.

Furthermore, this Element derives many original results and extends and reinterprets some well-known ones. For instance, it develops a concise and efficient expression for t.p.d.fs in the case of processes with stochastic volatility. Moreover, the analysis reveals an unexpected similarity between the propagation of vorticity in two-dimensional flows of viscous incompressible fluid and the motion of a harmonically bound particle, which is used to find a new explicit expression for the vorticity of a two-dimensional flow in terms of the Gaussian density.

Finally, the Element applies the new methodology to various financial engineering topics, such as pricing options with stochastic volatility, options with path-dependent volatility, Asian options, volatility and variance swaps, options on stocks with path-dependent volatility, and bonds and bond options. In contrast to the classical approach, the Element treats primary fixed-income products, such as bonds and bond options, as path-dependent, allowing us to gain additional intuition regarding such products’ pricing and risk management. It also highlights the flexibility of the interdisciplinary framework by incorporating additional complexities into the picture, such as jump-diffusion processes and, more generally, processes driven by affine pseudo-differential processes frequently used in financial applications.

1.3 Element Structure

Section 2 introduces Kelvin waves. Section 2.1 introduces the Euler equations, which describe the dynamics of a perfect fluid, alongside the Navier–Stokes equation for viscous incompressible fluids. Section 2.2 discusses the exact equilibria of these equations, focusing on states where velocity varies linearly and pressure quadratically with spatial coordinates, referred to as linear flows. Section 2.3 illustrates that the renowned Kelvin waves provide solutions to the linearized Euler and Navier–Stokes equations for small perturbations of the linear flows. This section also explores the use of Kelvin waves in analyzing the stability of these flows.

The Element uses Kelvin waves as a fundamental tool in the analytical arsenal, demonstrating their applicability across various study areas. For instance, they allow one to discover profound and surprising links between the viscous two-dimensional vorticity equations and the Klein–Kramers equation, a cornerstone of stochastic physics; see Section 6.6. This connection results in a novel formula representing vorticity as a Gaussian density and the stream function as the solution to the associated Poisson equation.

Section 3 investigates the degenerate stochastic process introduced by Kolmogorov in 1934, alongside the associated Fokker–Planck equation and its solution proposed by Kolmogorov. Further connections between the Kolmogorov and Klein–Kramers equations are explored in Section 4. To start with, Section 3 summarizes Kolmogorov’s original findings. Surprisingly, the Fokker–Planck equation, as used by Kolmogorov in his seminal paper, is inconsistent with his initial assumptions regarding the underlying process. Moreover, his proposed solution has dimensional inconsistencies and, as a result, does not satisfy the Fokker–Planck equation and initial conditions. However, there is a silver lining; Kolmogorov’s solution can be corrected via several complementary methods, which the section outlines. It concludes with an example of a representative corrected solution to the Kolmogorov problem.

Section 4 explores a selection of representative affine stochastic processes in statistical physics. First, it introduces the Langevin equation, which describes the dynamics of an underdamped Brownian particle in a potential field. Following this, it derives the Klein–Kramers equation, capturing the probabilistic aspects of the motion of such a particle. It turns out that the Kolmogorov equation derived in Section 3 is a particular case of the Klein–Kramers equation. The section presents Chandrasekhar’s solutions to the Klein–Kramers equations describing free and harmonically bound particles. The Klein–Kramers equation is inherently degenerate, with white noise impacting the particle’s velocity but not its position. It is shown in Section 8 that many path-dependent problems share this characteristic in mathematical finance. For instance, financial variables like the geometric price averages, which serve as the underlying instruments for a particular class of Asian options, can be conceptualized as path integrals, fitting into the category of degenerate stochastic processes.

Section 5 describes backward (Kolmogorov) and forward (Fokker–Planck) equations for t.p.d.fs of multidimensional stochastic jump-diffusion processes. The section explains the significance of studying t.p.d.fs. It sets up the general framework for Kolmogorov and Fokker–Planck equations and identifies the subset of affine stochastic processes amenable to analysis using the Kelvin-wave formalism. Subsequently, the section introduces an augmentation technique, providing a natural approach to tackle degenerate problems. Finally, it illustrates methods for transforming specific nonaffine processes into affine form through coordinate transformations, enhancing the scope of problems accessible by the Kelvin-wave methodology.

Section 6 studies Gaussian stochastic processes. It introduces a general formula for regular Gaussian processes, accommodating both degenerate scenarios and nondegenerate cases, as in Kolmogorov’s example. It expands this formula to address the practically significant scenario of killed Gaussian processes, followed by several illustrative examples. Then, the section presents the derivation of the t.p.d.f. for the Kolmogorov process with time-varying coefficients and explores the OU process with time-dependent coefficients and its extension, the augmented OU process, which models the combined dynamics of the process and its integral. Although the results are classical, their derivation through Kelvin-wave expansions provides a novel and enriching angle, offering an alternative viewpoint for understanding and deriving these established results. Next, the section examines free and harmonically bound particles, contrasting the Kelvin-wave method with Chandrasekhar’s classical approach. Finally, it revisits the basic concepts introduced in Section 2, demonstrating the akin nature of the temporal-spatial evolution of vorticity in the two-dimensional flow of a viscous fluid to the dynamics of a harmonically bound particle. This finding is intriguing and unexpected, forging a connection between seemingly unrelated physical phenomena.

Section 7 considers non-Gaussian processes. It starts with a general formula for non-Gaussian dynamics, accommodating degenerate and nondegenerate processes. Then, it expands this formula to killed processes. Several interesting examples are studied. These examples include a Kolmogorov process driven by anomalous diffusion, Feller processes with constant and time-dependent coefficients, and degenerate and nondegenerate augmented Feller processes. A novel method for investigating finite-time explosions of t.p.d.fs for augmented Feller processes is developed as a helpful by-product of the analysis. In addition, arithmetic Brownian motions with path-dependent volatility and degenerate and nondegenerate arithmetic Brownian motions with stochastic volatility are analyzed in detail.

Section 8 illustrates the application of the methodology to financial engineering. To start with, it lays the foundation of financial engineering, providing a primer for the uninitiated. Then, the section introduces the geometric Brownian motion, a staple in financial modeling, and discusses the modifications necessary to reflect the complexities of financial markets better. Several traditional models, such as Bachelier, Black–Scholes, Heston, and Stein–Stein models, and a novel path-dependent volatility model are explored via the Kelvin-wave formalism. In addition, it is shown how to price Asian options with geometric averaging via the Kolmogorov’s solution described in Section 3. Besides, volatility and variance swaps and swaptions, bonds and bond options are investigated by linking financial formulas to those used in physics for underdamped Brownian motion.

Section 9 succinctly outlines potential future expansions of the work presented in this Element and summarizes the conclusions. Finally, this Element is a revised and expanded version of Reference LiptonLipton (2023).

A note on notation: Given the wide-ranging scope of this Element, from hydrodynamics to molecular physics, probability theory, and financial engineering, adopting a unified notation system is impractical. Each field has its conventions carved in stone, leading to inevitable variations in notation. Notation is designed for consistency within and, where possible, across sections. However, readers are encouraged to remain vigilant to maintain coherence in their understanding.

2.1 Euler and Navier–Stokes Equations

Hydrodynamics studies how fluids (liquids and gases) move, primarily relying on fluid motion’s fundamental equations: the Euler and Navier–Stokes equations, with the Euler equations applicable to inviscid (frictionless) flow and the Navier–Stokes equations describing viscous fluids. Hydrodynamics has numerous applications across various fields, including engineering, astrophysics, oceanography, and climate change, among many others.

Recall that the Euler system of partial differential equations (PDEs) describing the motion of an inviscid, incompressible fluid has the form

$\begin{array}{rl}\frac{\mathrm{\partial }\mathbf{V}}{\mathrm{\partial }t}+(\mathbf{V}\cdot \mathrm{\nabla })\mathbf{V}+\mathrm{\nabla }\left(\frac{P}{\rho }\right)& =0,\\ \mathrm{\nabla }\cdot \mathbf{V}& =0;\end{array}$(2.1)

where $t$ is time, $\mathbf{x}$ is the position, $V\left(t,\mathbf{x}\right)$ is the velocity vector, $P\left(t,\mathbf{x}\right)$ is the pressure, $\rho$ is the constant density, $\mathrm{\nabla }$ is the gradient, and $\cdot$ denotes the scalar product; see, for example, Reference ChandrasekharChandrasekhar (1961). In Cartesian coordinates, the equations in (2.1) can be written as follows:

$\begin{array}{rl}\frac{\mathrm{\partial }{V}_i}{\mathrm{\partial }t}+V_j\frac{\mathrm{\partial }{V}_i}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}\left(\frac{P}{\rho }\right)& =0,\\ \frac{\mathrm{\partial }{V}_i}{\mathrm{\partial }{x}_i}& =0.\end{array}$(2.2)

Here and in what follows, Einstein’s summation convention over repeated indices is used.

The motion of the incompressible viscous fluid is described by the classical Navier–Stokes equations of the form:

$\begin{array}{r}\frac{\mathrm{\partial }\mathbf{V}}{\mathrm{\partial }t}+(\mathbf{V}\cdot \mathrm{\nabla })\mathbf{V}-\nu \mathrm{\Delta }\mathbf{V}\mathbf{+}\mathrm{\nabla }\left(\frac{P}{\rho }\right)=0,\\ \mathrm{\nabla }\cdot \mathbf{V}=0;\end{array}$(2.3)

where $\nu$ is the kinematic viscosity; see, for example, Reference ChandrasekharChandrasekhar (1961). Explicitly,

$\begin{array}{r}\frac{\mathrm{\partial }{V}_i}{\mathrm{\partial }t}+V_j\frac{\mathrm{\partial }{V}_i}{\mathrm{\partial }{x}_j}-\nu \frac{{\mathrm{\partial }}^2{V}_i}{\mathrm{\partial }{x}_j\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}\left(\frac{P}{\rho }\right)=0,\\ \frac{\mathrm{\partial }{V}_i}{\mathrm{\partial }{x}_i}=0.\end{array}$(2.4)

The diffusive term $\mathbf{-}\nu \mathrm{\Delta }\mathbf{V}$ in (2.4) describes frictions ignored in (2.3). Due to their greater generality, the Navier–Stokes equations are fundamental to understanding important phenomena, such as the transition from laminar to turbulent flow.

2.2 Linear Flows

This section studies exact solutions of the Euler and Navier–Stokes equations known as linear flows. These solutions are valuable for several reasons: (a) exact solutions provide precise, analytical descriptions of fluid flow patterns under specific conditions; (b) they serve as benchmarks for understanding fundamental hydrodynamics phenomena like wave propagation; (c) they provide a bridge which is crucial for more complex studies by simplifying the inherently complex and nonlinear nature of hydrodynamics, and making it possible to understand the behavior of more general fluid flows. Linear solutions of the Euler and Navier–Stokes equations help to study fluid flow stability. This understanding is crucial in predicting and controlling flow behavior in various engineering applications, from aerospace to hydraulic engineering. By starting with linear solutions, one can incrementally introduce nonlinear effects, allowing for a systematic study of nonlinear phenomena in hydrodynamics. This approach can uncover the mechanisms behind complex flows, including turbulence and chaotic flow behaviors. Exact linear solutions of the Euler equations provide a clear, analytical framework for exploring the behavior of fluids and validating more complicated models.

It is easy to show that the equations in (2.1) have a family of solutions $\left(\mathbf{V}\left(t,\mathbf{x}\right),P\left(t,\mathbf{x}\right)\right),$ linearly depending on spatial coordinates:

$\begin{array}{r}\mathbf{V}\left(t,\mathbf{x}\right)=\mathfrak{L}\left(t\right)\mathbf{x}\mathbf{,}\frac{P\left(t,\mathbf{x}\right)}{\rho }=\frac{P_0}{\rho }+\frac{1}{2}\mathfrak{M}\left(t\right)\mathbf{x}\cdot \mathbf{x}\mathbf{,}\end{array}$(2.5)

where the $3\times 3$ matrices $\mathfrak{L}\left(t\right),$ $\mathfrak{M}\left(t\right),$ are such that

$\begin{array}{rl}\frac{d\mathfrak{L}\left(t\right)}{dt}+{\mathfrak{L}}^2\left(t\right)+\mathfrak{M}\left(t\right)& =0,\\ \mathrm{T}\mathrm{r}\left(\mathfrak{L}\left(t\right)\right)=0,\mathfrak{M}\left(t\right)& ={\mathfrak{M}}^{*}\left(t\right).\end{array}$(2.6)

It is clear that linear flows, given by (2.5), are unaffected by viscosity, hence they satisfy (2.14).

Flows (2.5) have stagnation points at the origin. Typical examples are planar flows of the form

$\begin{array}{rl}{V}_1& =\frac{1}{2}\left(s{x}_1-w{x}_2\right),V_2=\frac{1}{2}\left(w{x}_1-s{x}_2\right),V_3=0,\\ \frac{P}{\rho }& =\frac{P_0}{\rho }+\frac{1}{4}\left(w^2-s^2\right)\left(x_1^2+x_2^2\right).\end{array}$(2.7)

These flows are elliptic when $s<w,$ and hyperbolic otherwise; see, for example, Reference Friedlander and Lipton-LifschitzFriedlander and Lipton-Lifschitz (2003).

2.3 Kelvin Waves in an Incompressible Fluid

The study of small perturbations of exact solutions of the Euler and Navier–Stokes equations is the core of the stability analysis in fluid dynamics. Examining their behavior is essential for predicting how fluid flows evolve under slight disturbances. One can determine whether a particular flow is stable or unstable by introducing small perturbations to an exact solution and observing the system’s response. If these perturbations grow over time, the flow is considered unstable; if they decay or remain bounded, the flow is stable. One of this analysis’s most critical applications is understanding the transition from laminar (smooth and orderly) to turbulent (chaotic and unpredictable) flows. Small perturbations can exhibit exponential growth, leading to the onset of turbulence. For more detailed investigations, direct numerical simulations of the perturbed Navier–Stokes equations can be used to study the nonlinear evolution of perturbations. This approach can capture the complete transition from initial instability to fully developed turbulence, offering insights into the complex interactions that drive flow dynamics. The study of perturbations offers theoretical insights into the fundamental nature of fluid dynamics, including the mechanisms of flow instability, transition, and turbulence structure. It helps in developing reduced-order models and theories that explain complex fluid phenomena. Here, Kelvin waves are used as the primary tool for studying small perturbations of linear flows. In the rest of this Element, Kelvin waves are used for other purposes. This section is dedicated to their brief description.

It is necessary to study the behavior of perturbations of solutions given by (2.5), which are denoted by $\left(\mathbf{v}\left(t,\mathbf{x}\right),p\left(t,\mathbf{x}\right)\right).$ By neglecting the quadratic term ($v\cdot \mathrm{\nabla })v,$ one can write the system of PDEs for $\left(\mathbf{v}\mathbf{,}p\right)$ as follows:

$\begin{array}{rl}\frac{\mathrm{\partial }\mathbf{v}}{\mathrm{\partial }t}+(\mathfrak{L}\left(t\right)\mathbf{x}\cdot \mathrm{\nabla })\mathbf{v}\mathbf{+}\mathfrak{L}\left(t\right)\mathbf{v}\mathbf{+}\mathrm{\nabla }\left(\frac{p}{\rho }\right)& =0,\\ \mathrm{\nabla }\cdot \mathbf{v}& =0.\end{array}$(2.8)

It has been known for a long time that linear PDEs (2.8) have wavelike solutions of the form:

$\begin{array}{r}\left(\mathbf{v}\left(t,\mathbf{x}\right),\frac{p\left(t,\mathbf{x}\right)}{\rho }\right)=\left(\mathbf{a}\left(t\right),a\left(t\right)\right)exp\left(i\mathit{\beta }\left(t\right)\cdot \left(\mathbf{x}\mathbf{-}\mathbf{r}\left(t\right)\right)\right),\end{array}$(2.9)

where $\left(\mathbf{a}\left(t\right),a\left(t\right)\right)$ are time-dependent amplitudes, and $\mathit{\beta }\left(t\right)$ is the time-dependent wave vector; see Reference KelvinKelvin (1887), Reference OrrOrr (1907), Reference Craik and CriminaleCraik and Criminale (1986), and Reference Friedlander and Lipton-LifschitzFriedlander and Lipton-Lifschitz (2003). In this Element, these solutions are called the Kelvin waves. It should be emphasized that the so-called affine ansatz is a special instance of Kelvin wave. This observation allows one to discover similarities among seemingly unrelated topics, which, in turn, facilitates their holistic and comprehensive study. An excerpt from Kelvin’s original paper is shown in Figure 2.

Figure 2 An excerpt from Kelvin’s original paper, where Kelvin waves are introduced for the first time; see Reference KelvinKelvin (1887). Public domain.

As one can see from Figure 2, Kelvin considered the special case of the so-called shear linear flow of the form

$\begin{array}{r}\mathbf{V}\left(t,\mathbf{x}\right)=\left(V_1\left(x_2\right),0,0\right)=\left(l_{12}{x}_2,0,0\right),\end{array}$(2.10)

between two plates, $x_2=0$ and $x_2=L,$ the first one at rest and the second one moving in parallel.

The triplet $\mathbf{r}\left(t\right),$ $\mathit{\beta }\left(t\right),$ $\mathbf{a}\left(t\right)$ satisfies the following system of ODEs:

$\begin{array}{rl}\frac{d\mathbf{r}\left(t\right)}{dt}-\mathfrak{L}\left(t\right)\mathbf{r}\left(t\right)=0,\mathbf{r}\left(0\right)& ={\mathbf{r}}_0,\\ \frac{d\mathit{\beta }\left(t\right)}{dt}+{\mathfrak{L}}^{*}\left(t\right)\mathit{\beta }\left(t\right)=0,\mathit{\beta }\left(0\right)& ={\mathit{\beta }}_0,\\ \frac{d\mathbf{a}\left(t\right)}{dt}+\mathfrak{L}\left(t\right)\mathbf{a}\left(t\right)-2\frac{\mathfrak{L}\left(t\right)\mathbf{a}\left(t\right)\cdot \mathit{\beta }\left(t\right)}{\mathit{\beta }\left(t\right)\cdot \mathit{\beta }\left(t\right)}\mathit{\beta }\left(t\right)=0,\mathbf{a}\left(0\right)& ={\mathbf{a}}_0,\\ {\mathit{\beta }}_0\cdot {\mathbf{a}}_0& =0.\end{array}$(2.11)

Here and in what follows, the superscript $*$ stands for transpose. The corresponding $p\left(t\right)$ can be found via the incompressibility condition. It is easy to show that for $t\ge 0,$

$\begin{array}{r}\mathit{\beta }\left(t\right)\cdot \mathbf{r}\left(t\right)={\mathit{\beta }}_0\cdot {\mathbf{r}}_0,\mathit{\beta }\left(t\right)\cdot \mathbf{a}\left(t\right)=0.\end{array}$(2.12)

Thus, the Kelvin-wave formalism results in ingenious separation of variables and allows us to solve a system of ODEs (2.11), rather than PDEs (2.8).

Typically, the equations in (2.11) are used to study the stability of the linear flow. Such a flow is unstable whenever $∥\mathbf{a}\left(t\right)∥\to \mathrm{\infty }$ for some choices of ${\mathit{\beta }}_0,{\mathbf{a}}_0$; see Reference BaylyBayly (1986), Reference LifschitzLifschitz (1995), and Reference Bayly, Holm and LifschitzBayly et al. (1996). Moreover, it can be shown that the same instabilities occur in general three-dimensional flows, because locally they are equivalent to linear flows; see Reference Lifschitz and HameiriLifschitz and Hameiri (1991a), Reference Friedlander and VishikFriedlander and Vishik (1991), Reference Lifschitz and HameiriLifschitz and Hameiri (1991b), and Reference Friedlander and Lipton-LifschitzFriedlander and Lipton-Lifschitz (2003).

Interestingly, Reference ChandrasekharChandrasekhar (1961) pointed out that the superposition of the linear flow (2.5) and the Kelvin wave (2.9), namely,

$\begin{array}{rl}\tilde{\mathbf{V}}\left(t,\mathbf{x}\right)& =\mathfrak{L}\left(t\right)\mathbf{x}\mathbf{+}\mathbf{v}\left(t,\mathbf{x}\right),\\ \frac{\tilde{P}\left(t,\mathbf{x}\right)}{\rho }& =\frac{1}{2}\mathfrak{M}\left(t\right)\mathbf{x}\cdot \mathbf{x}\mathbf{+}\frac{p\left(t,\mathbf{x}\right)}{\rho },\end{array}$(2.13)

satisfies the nonlinear Euler equations (2.1) since the nonlinear term $(\mathbf{v}\cdot \mathrm{\nabla })\mathbf{v}$ vanishes identically due to incompressibility.Footnote ³ Studying secondary instabilities of flows with elliptic streamlines, that is, instabilities of Kelvin waves is an important and intricate topic; see Reference Fabijonas, Holm and LifschitzFabijonas et al. (1997).

Viscosity does affect small perturbations of linear flows. For viscous incompressible fluids, Kelvin waves are governed by the following equations:

$\begin{array}{rl}\frac{\mathrm{\partial }\mathbf{v}}{\mathrm{\partial }t}+(\mathfrak{L}\left(t\right)\mathbf{x}\cdot \mathrm{\nabla })\mathbf{v}\mathbf{+}\mathfrak{L}\left(t\right)\mathbf{v}-\nu \mathrm{\Delta }\mathbf{v}\mathbf{+}\mathrm{\nabla }\left(\frac{p}{\rho }\right)& =0,\\ \mathrm{\nabla }\cdot \mathbf{v}& =0.\end{array}$(2.14)

The viscous version of (2.11) has the following form; see Reference LifschitzLifschitz (1991):

$\begin{array}{rl}& \frac{d\mathbf{r}\left(t\right)}{dt}-\mathfrak{L}\left(t\right)\mathbf{r}\left(t\right)=0,\mathbf{r}\left(0\right)={\mathbf{r}}_0,\\ & \frac{d\mathit{\beta }\left(t\right)}{dt}+{\mathfrak{L}}^{*}\left(t\right)\mathit{\beta }\left(t\right)=0,\mathit{\beta }\left(0\right)={\mathit{\beta }}_0,\\ & \frac{d\mathbf{a}\left(t\right)}{dt}+\mathfrak{L}\left(t\right)\mathbf{a}\left(t\right)-2\frac{\mathfrak{L}\left(t\right)\mathbf{a}\left(t\right)\cdot \mathit{\beta }\left(t\right)}{\mathit{\beta }\left(t\right)\cdot \mathit{\beta }\left(t\right)}\mathit{\beta }\left(t\right)+\nu {\left|\mathit{\beta }\left(t\right)\right|}^2\mathbf{a}\left(t\right)=0,\mathbf{a}\left(0\right)={\mathbf{a}}_0,\\ & {\mathit{\beta }}_0\cdot {\mathbf{a}}_0=0.\end{array}$(2.15)

It is shown in Section 6.5 that in the two-dimensional case, the Navier–Stokes equations for small perturbations of linear flows are more or less identical to the Fokker–Planck equations for harmonically bound articles, which is surprising.

The evolution of a typical Kelvin wave parameters triplet $\mathbf{r}\left(t\right),$ $\mathit{\beta }\left(t\right),$ $\mathbf{a}\left(t\right)$ is illustrated in Figure 3. The impact of viscosity is illustrated in Figure 4. These figures show that depending on the initial orientation of the wave vector $\mathit{\beta }\left(t\right),$ the amplitude $\mathbf{a}\left(t\right)$ can be either bounded or unbounded. For elliptic flows, unbounded amplitudes are always present for specific orientations, so all of them are unstable; see Reference BaylyBayly (1986), Reference Bayly, Holm and LifschitzBayly et al. (1996), Reference Friedlander and Lipton-LifschitzFriedlander and Lipton-Lifschitz (2003), and references therein.

Figure 3 Kelvin waves corresponding to two different orientations of the initial wave vector $\beta \left(0\right)$ and $\mathbf{a}\left(0\right).$ (a), (b) $\beta (0)=\left(sin\left(\pi /4\right),0,cos\left(\pi /4\right)\right),$ $\mathbf{a}\left(0\right)=\left(0,sin\left(\pi /4\right),0\right)$; (c), (d) $\beta (0)=\left(sin\left(\pi /3\right),0,cos\left(\pi /3\right)\right),$ $\mathbf{a}\left(0\right)=\left(0,sin\left(\pi /3\right),0\right).$ Other parameters are as follows: $T=100,$ $\omega =1,$ $s=0.5.$ In the first case, $\mathbf{a}\left(t\right)$ stays bounded, while $\mathbf{a}\left(t\right)$ explodes in the second case. This explosion means that the underlying elliptic flow is unstable. Author’s graphics.

Figure 4 Kelvin waves in the viscous fluid with viscosity $\nu =0.07.$ Other parameters and initial conditions are the same as in Figure 3. Viscosity dampens the instability but, generally, does not suppress it entirely. Author’s graphics.

3.1 Background

The Kolmogorov equation studies the evolution of a particle in the phase space. The particle’s position and velocity evolve in time due to the interplay between the deterministic drift and stochastic force affecting only its velocity. Since only the particle’s velocity is affected by the random force, the PDE describing the evolution of the t.p.d.f. in the phase space is degenerate. The Kolmogorov equation is a particular case of the Klein–Kramers equation studied in Section 4.

The significance of the Kolmogorov equation lies in its ability to model the intricate balance between deterministic behavior and stochastic dynamics, providing a basic framework for studying the evolution of systems in phase space. It has important applications in various fields, including physics for understanding particle dynamics, finance for modeling asset prices, and beyond. It demonstrates the profound interplay between stochastic processes and differential equations.

The Kolmogorov equation is hypoelliptic; as such, it serves as a prototype for a broad class of hypoelliptic PDEs. Although it does not meet the exact criteria for ellipticity (due to the second-order derivatives not being present in all directions of the phase space), the solutions to the equation are still smooth, which is particularly important in the context of stochastic processes, where hypoellipticity ensures that the probability density function remains smooth and well-behaved, facilitating the analysis of the system’s dynamics over time.

3.2 Summary of Kolmogorov’s Paper

In a remarkable (and remarkably concise) note, Kolmogorov considers a system of particles in $n$-dimensional space with coordinates $q_1,\dots ,q_n,$ and velocities ${\dot{q}}_1,\dots ,{\dot{q}}_n,$ assuming the probability density function

$g\left(t,q_1,\dots ,q_n,{\dot{q}}_1,\dots ,{\dot{q}}_n,t^{\mathrm{\prime }},q_1^{\mathrm{\prime }},\dots ,q_n^{\mathrm{\prime }},{\dot{q}}_1^{\mathrm{\prime }},\dots ,{\dot{q}}_n^{\mathrm{\prime }}\right)$

exists for some time $t^{\mathrm{\prime }}>t,$ and reveals (without any explanation) an analytical expression for $g$ in the one-dimensional case; see Reference KolmogoroffKolmogoroff (1934).Footnote ⁴ This note is the third in a series of papers, the previous two being Reference KolmogoroffKolmogoroff (1931), Reference Kolmogoroff(1933).

Kolmogorov makes the following natural assumptions:

$\begin{array}{rl}\mathbf{E}\left|\mathrm{\Delta }{q}_i-{\dot{q}}_i\mathrm{\Delta }t\right|& =o\left(\mathrm{\Delta }t\right),\end{array}$(3.1)

$\begin{array}{rl}\mathbf{E}{\left(\mathrm{\Delta }{q}_i\right)}^2& =o\left(\mathrm{\Delta }t\right),\end{array}$(3.2)

where $\mathrm{\Delta }t=t^{\mathrm{\prime }}-t.$ Equations (3.1) and (3.2) imply

$\begin{array}{rl}& \mathbf{E}\left(\mathrm{\Delta }{q}_i\right)={\dot{q}}_i\mathrm{\Delta }t+o\left(\mathrm{\Delta }t\right),\end{array}$(3.3)

$\begin{array}{rl}& \mathbf{E}\left(\mathrm{\Delta }{q}_i\mathrm{\Delta }{q}_j\right)\le \sqrt{\mathbf{E}{\left(\mathrm{\Delta }{q}_i\right)}^2\mathbf{E}{\left(\mathrm{\Delta }{q}_j\right)}^2}=o\left(\mathrm{\Delta }t\right).\end{array}$(3.4)

Furthermore, under very general assumptions, the following relationships hold:

$\begin{array}{rl}\mathbf{E}\left(\mathrm{\Delta }{\dot{q}}_i\right)& =f_i\left(t,q,\dot{q}\right)\mathrm{\Delta }t+o\left(\mathrm{\Delta }t\right),\end{array}$(3.5)

$\begin{array}{rl}\mathbf{E}{\left(\mathrm{\Delta }{\dot{q}}_i\right)}^2& =k_{ii}\left(t,q,\dot{q}\right)\mathrm{\Delta }t+o\left(\mathrm{\Delta }t\right),\end{array}$(3.6)

$\begin{array}{rl}\mathbf{E}\left(\mathrm{\Delta }{\dot{q}}_i\mathrm{\Delta }{\dot{q}}_j\right)& =k_{ij}\left(t,q,\dot{q}\right)\mathrm{\Delta }t+o\left(\mathrm{\Delta }t\right),\end{array}$(3.7)

where $f$ and $k$ are continuous functions. Equations (3.2), and (3.6) imply

$\begin{array}{r}\mathbf{E}\left(\mathrm{\Delta }{\dot{q}}_i\mathrm{\Delta }{\dot{q}}_j\right)\le \sqrt{\mathbf{E}{\left(\mathrm{\Delta }{\dot{q}}_i\right)}^2\mathbf{E}{\left(\mathrm{\Delta }{\dot{q}}_j\right)}^2}=o\left(\mathrm{\Delta }t\right).\end{array}$(3.8)

Under some natural physical assumptions, it follows that $g$ satisfies the following differential equation of the Fokker–Planck type:

$\begin{array}{r}\frac{\mathrm{\partial }g}{\mathrm{\partial }{t}^{\mathrm{\prime }}}=-\sum {\dot{q}}_i^{\mathrm{\prime }}\frac{\mathrm{\partial }g}{\mathrm{\partial }{q}_i^{\mathrm{\prime }}}-\sum \frac{\mathrm{\partial }}{\mathrm{\partial }{\dot{q}}_i^{\mathrm{\prime }}}\left\{f_i\left(t,q,\dot{q}\right)g\right\}+\sum \sum \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\dot{q}}_i^{\mathrm{\prime }}\mathrm{\partial }{\dot{q}}_j^{\mathrm{\prime }}}\left\{k\left(t,q,\dot{q}\right)g\right\}.\end{array}$(3.9)

In the one-dimensional case, one has

$\begin{array}{r}\frac{\mathrm{\partial }g}{\mathrm{\partial }{t}^{\mathrm{\prime }}}=-{\dot{q}}^{\mathrm{\prime }}\frac{\mathrm{\partial }g}{\mathrm{\partial }{q}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }}{\mathrm{\partial }{\dot{q}}^{\mathrm{\prime }}}\left\{f\left(t,q,\dot{q}\right)g\right\}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\dot{q}}^{\mathrm{\prime }2}}\left\{k\left(t,q,\dot{q}\right)g\right\}.\end{array}$(3.10)

These equations are known as ultra-parabolic Fokker–Plank–Kolmogorov equations due to their degeneracy.

When $f$ and $k$ are constants, (3.10) becomes

$\begin{array}{r}\frac{\mathrm{\partial }g}{\mathrm{\partial }{t}^{\mathrm{\prime }}}=-{\dot{q}}^{\mathrm{\prime }}\frac{\mathrm{\partial }g}{\mathrm{\partial }{q}^{\mathrm{\prime }}}-f\frac{\mathrm{\partial }g}{\mathrm{\partial }{\dot{q}}^{\mathrm{\prime }}}+k\frac{{\mathrm{\partial }}^{2}g}{\mathrm{\partial }{\dot{q}}^{\mathrm{\prime }2}}.\end{array}$(3.11)

The corresponding fundamental solution of has the following form:

$\begin{array}{r}g=\frac{2\sqrt{3}}{\pi k^2{\left(t^{\mathrm{\prime }}-t\right)}^2}exp\left\{-\frac{{\left({\dot{q}}^{\mathrm{\prime }}-\dot{q}-f\left(t^{\mathrm{\prime }}-t\right)\right)}^2}{4k\left(t^{\mathrm{\prime }}-t\right)}-\frac{3{\left(q^{\mathrm{\prime }}-q-\frac{{\dot{q}}^{\mathrm{\prime }}+\dot{q}}{2}\left(t^{\mathrm{\prime }}-t\right)\right)}^2}{k^3{\left(t^{\mathrm{\prime }}-t\right)}^3}\right\}.\end{array}$(3.12)

One can see that $\mathrm{\Delta }\dot{q}$ is of the order ${\left(\mathrm{\Delta }t\right)}^{1/2}.$ At the same time

$\begin{array}{r}\mathrm{\Delta }q=\dot{q}\mathrm{\Delta }t+O{\left(\mathrm{\Delta }t\right)}^{3/2}.\end{array}$(3.13)

One can prove that a similar relation holds for the general (3.9).

Kolmogorov’s original paper is shown in Figure 5.

Reproduced by kind permission of the Editors of Annals of Mathematics.

Figure 5 Kolmogorov’s original paper, presented here for the inquisitive reader to enjoy; see Reference KolmogoroffKolmogoroff (1934).

Kolmogorov equations fascinated mathematicians for a long time and generated a great deal of research; see, for example, Reference WeberWeber (1951), Reference HörmanderHörmander (1967), Reference KuptsovKuptsov (1972), Reference Lanconelli, Pascucci, Polidoro, Sh, Birman, Solonnikov and UraltsevaLanconelli et al. (2002), Reference Pascucci and Parabolic ProblemsPascucci (2005), Reference Ivasishen and MedynskyIvasishen and Medynsky (2010), and Reference Duong and TranDuong & Tran (2018), among others.

It is worth mentioning that physicists derived Equations (3.9) and (3.10) at least a decade earlier than Kolmogorov; see Section 4.

3.3 Challenge and Response

Despite its undoubted brilliance, Kolmogorov’s original paper has several issues.

First, Equations (3.9) and (3.10) are not the Fokker–Planck equations associated with the process described by Equations (3.5)–(3.7), since they lack the prefactor $1/2$ in front of the diffusion terms. The corrected multivariate equation has the form

$\begin{array}{rl}\frac{\mathrm{\partial }g}{\mathrm{\partial }{t}^{\mathrm{\prime }}}=& -\sum {\dot{q}}_i^{\mathrm{\prime }}\frac{\mathrm{\partial }}{\mathrm{\partial }{q}_i^{\mathrm{\prime }}}g-\sum \frac{\mathrm{\partial }}{\mathrm{\partial }{\dot{q}}_i^{\mathrm{\prime }}}\left\{f_i\left(t,q,\dot{q}\right)g\right\}\\ & +\frac{1}{2}\sum \sum \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\dot{q}}_i^{\mathrm{\prime }}\mathrm{\partial }{\dot{q}}_j^{\mathrm{\prime }}}\left\{\left\{k\left(t,q,\dot{q}\right)g\right\}\right\},\end{array}$(3.14)

while the corresponding one-dimensional equation has the form

$\begin{array}{r}\frac{\mathrm{\partial }g}{\mathrm{\partial }{t}^{\mathrm{\prime }}}=-{\dot{q}}^{\mathrm{\prime }}\frac{\mathrm{\partial }}{\mathrm{\partial }{q}^{\mathrm{\prime }}}g-\frac{\mathrm{\partial }}{\mathrm{\partial }{\dot{q}}^{\mathrm{\prime }}}\left\{f\left(t,q,\dot{q}\right)g\right\}+\frac{1}{2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\dot{q}}^{\mathrm{\prime }2}}\left\{\left\{k\left(t,q,\dot{q}\right)g\right\}\right\}.\end{array}$(3.15)

Alternatively, Equations (3.6) and (3.7) can be altered as follows:

$\begin{array}{rl}\mathbf{E}{\left(\mathrm{\Delta }{\dot{q}}_i\right)}^2& =2k_{ii}\left(t,q,\dot{q}\right)\mathrm{\Delta }t+o\left(\mathrm{\Delta }t\right),\end{array}$(3.16)

$\begin{array}{rl}\mathbf{E}\left(\mathrm{\Delta }{\dot{q}}_i\mathrm{\Delta }{\dot{q}}_j\right)& =2k_{ij}\left(t,q,\dot{q}\right)\mathrm{\Delta }t+o\left(\mathrm{\Delta }t\right).\end{array}$(3.17)

In the following discussion, the Fokker–Planck equation is updated.

Second, $g$ given by (3.12) does not solve (3.10). It also does not satisfy the (implicit) initial condition

$\begin{array}{r}g\left(t,q,\dot{q},t,q^{\mathrm{\prime }},{\dot{q}}^{\mathrm{\prime }}\right)=\delta \left(q^{\mathrm{\prime }}-q\right)\delta \left({\dot{q}}^{\mathrm{\prime }}-\dot{q}\right),\end{array}$(3.18)

where $\delta \left(.\right)$ is the Dirac $\delta$-function. The fact that expression (3.12) does not solve (3.10) can be verified by substitution. However, it is easier to verify this statement via dimensional analysis. The dimensions of the corresponding variables and coefficients are as follows:

$\begin{array}{rl}& \left[t\right]=\left[t^{\mathrm{\prime }}\right]=T,\left[q\right]=\left[q^{\mathrm{\prime }}\right]=L,\left[\dot{q}\right]=\left[{\dot{q}}^{\mathrm{\prime }}\right]=\frac{L}{T},\left[g\right]=\frac{T}{L^2},\left[f\right]=\frac{L}{T^2},\\ & \left[k\right]=\frac{L^2}{T^3}.\end{array}$(3.19)

It is easy to show that $g$ is scale-invariant, so that

$\begin{array}{r}g\left({\lambda }^{2}t,{\lambda }^{3}q,\lambda \dot{q},{\lambda }^2{t}^{\mathrm{\prime }},{\lambda }^3{q}^{\mathrm{\prime }},\lambda {\dot{q}}^{\mathrm{\prime }};{\lambda }^{-1}f,k\right)={\lambda }^{-4}g\left(t,q,\dot{q},t^{\mathrm{\prime }},q^{\mathrm{\prime }},{\dot{q}}^{\mathrm{\prime }};f,k\right).\end{array}$(3.20)

The original Kolmogorov formula contains two typos, making it dimensionally incorrect since the term

$\begin{array}{r}\frac{3{\left(q^{\mathrm{\prime }}-q-\frac{{\dot{q}}^{\mathrm{\prime }}+\dot{q}}{2}\left(t^{\mathrm{\prime }}-t\right)\right)}^2}{k^3{\left(t^{\mathrm{\prime }}-t\right)}^3}\end{array}$

in the exponent is not nondimensional, as it should be, and has dimension $T^6{L}^{-1},$ while the prefactor

$\begin{array}{r}\frac{2\sqrt{3}}{\pi k^2{\left(t^{\mathrm{\prime }}-t\right)}^2}\end{array}$

has dimension $T^4{L}^{-1},$ instead of the right dimension, $T{L}^{-2}.$

Third, due to yet another typo, the solution given by (3.12) does not converge to the initial condition in the limit $t^{\mathrm{\prime }}\to t.$ Indeed, asymptotically, one has

$\begin{array}{r}g\sim H\left(\frac{k^3{\left(t^{\mathrm{\prime }}-t\right)}^3}{6},q^{\mathrm{\prime }}-q\right)H\left(2k\left(t^{\mathrm{\prime }}-t\right),{\dot{q}}^{\mathrm{\prime }}-\dot{q}\right)\to 4\delta \left(q^{\mathrm{\prime }}-q\right)\delta \left({\dot{q}}^{\mathrm{\prime }}-\dot{q}\right),\end{array}$(3.21)

where $H\left(\mu ,\nu \right)$ is the standard heat kernel:

$\begin{array}{r}H\left(\mu ,\nu \right)=\frac{exp\left(-\frac{{\nu }^2}{2\mu }\right)}{\sqrt{2\pi \mu }}.\end{array}$(3.22)

However, not all is lost. Dimensional analysis shows that the correct solution $g\left(t,q,\dot{q},t^{\mathrm{\prime }},q^{\mathrm{\prime }},{\dot{q}}^{\mathrm{\prime }};f,k\right)$ of (3.10) has the following form:

$\begin{array}{r}g=\frac{\sqrt{3}}{2\pi k{\left(t^{\mathrm{\prime }}-t\right)}^2}exp\left\{-\frac{{\left({\dot{q}}^{\mathrm{\prime }}-\dot{q}-f\left(t^{\mathrm{\prime }}-t\right)\right)}^2}{4k\left(t^{\mathrm{\prime }}-t\right)}-\frac{3{\left(q^{\mathrm{\prime }}-q-\frac{{\dot{q}}^{\mathrm{\prime }}+\dot{q}}{2}\left(t^{\mathrm{\prime }}-t\right)\right)}^2}{k{\left(t^{\mathrm{\prime }}-t\right)}^3}\right\},\end{array}$(3.23)

which is not far from Kolmogorov’s formula. Similarly, the correct solution of (3.15) has the following form:

$\begin{array}{r}g=\frac{\sqrt{3}}{\pi k{\left(t^{\mathrm{\prime }}-t\right)}^2}exp\left\{-\frac{{\left({\dot{q}}^{\mathrm{\prime }}-\dot{q}-f\left(t^{\mathrm{\prime }}-t\right)\right)}^2}{2k\left(t^{\mathrm{\prime }}-t\right)}-\frac{6{\left(q^{\mathrm{\prime }}-q-\frac{{\dot{q}}^{\mathrm{\prime }}+\dot{q}}{2}\left(t^{\mathrm{\prime }}-t\right)\right)}^2}{k{\left(t^{\mathrm{\prime }}-t\right)}^3}\right\}.\end{array}$(3.24)

3.4 Direct Verification

In order to avoid confusion, from now on, the notation is changed to make the formulas easier to read. Specifically, it is assumed that $\bar{x}$ represents the position of a particle at time $\bar{t}$ and $x$ its position at time $t,$ while $\bar{y}$ represents its velocity at time $\bar{t},$ and $y$ its velocity at time $t,$ so that

$\begin{array}{r}\left(t,q,\dot{q}\right)\to \left(t,x,y\right),\left(t^{\mathrm{\prime }},q^{\mathrm{\prime }},{\dot{q}}^{\mathrm{\prime }}\right)\to \left(\bar{t},\bar{x},\bar{y}\right).\end{array}$(3.25)

One of our objectives is deriving the (corrected) Kolmogorov formula from first principles using Kelvin waves. Subsequently, it is shown how to use it in the financial mathematics context. The governing SDE can be written as

$\begin{array}{r}d{\hat{x}}_t={\hat{y}}_{t}dt,{\hat{x}}_t=x,\\ d{\hat{y}}_t=bdt+\sigma d{\hat{W}}_t,{\hat{y}}_t=y.\end{array}$(3.26)

The corresponding Fokker–Planck–Kolmogorov problem for the t.p.d.f. $\varpi \left(t,x,y,\bar{t},\bar{x},\bar{y}\right)$ has the form:

$\begin{array}{rl}{\varpi }_{\bar{t}}& \left(t,x,y,\bar{t},\bar{x},\bar{y}\right)-\frac{1}{2}{\sigma }^2{\varpi }_{\bar{y}\bar{y}}\left(t,x,y,\bar{t},\bar{x},\bar{y}\right)\\ & +\bar{y}{\varpi }_{\bar{x}}\left(t,x,y,\bar{t},\bar{x},\bar{y}\right)+b{\varpi }_{\bar{y}}\left(t,x,y,\bar{t},\bar{x},\bar{y}\right)=0,\\ & \varpi \left(t,x,y,t,\bar{x},\bar{y}\right)=\delta \left(\bar{x}-x\right)\delta \left(\bar{y}-y\right).\end{array}$(3.27)

The solution of (3.27) is as follows:

$\begin{array}{r}\varpi \left(t,x,y,\bar{t},\bar{x},\bar{y}\right)=\frac{\sqrt{3}}{\pi {\sigma }^2{T}^2}exp\left(-\mathrm{\Phi }\left(t,x,y,\bar{t},\bar{x},\bar{y}\right)\right),\end{array}$(3.28)

where

$\begin{array}{r}\mathrm{\Phi }\left(t,x,y,\bar{t},\bar{x},\bar{y}\right)=\frac{{\left(\bar{y}-y-bT\right)}^2}{2{\sigma }^{2}T}+\frac{6{\left(\bar{x}-x-\frac{\left(\bar{y}+y\right)T}{2}\right)}^2}{{\sigma }^2{T}^3}=\frac{A^2}{2}+6B^2,\end{array}$(3.29)

and

$\begin{array}{r}A=\frac{\left(\bar{y}-y-bT\right)}{\sqrt{{\sigma }^{2}T}},\left[A\right]=1,B=\frac{\left(\bar{x}-x-\frac{\left(\bar{y}+y\right)T}{2}\right)}{\sqrt{{\sigma }^2{T}^3}},\left[B\right]=1.\end{array}$(3.30)

Here and in what follows, the following shorthand notation is used:

$\begin{array}{r}T=\bar{t}-t.\end{array}$(3.31)

Let us check that $\varpi$ satisfies the Fokker–Planck equation and the initial conditions. A simple calculation yields:

$\begin{array}{rl}{\mathrm{\Phi }}_{\bar{t}}& =-\left(\frac{A^2}{2T}+\frac{bA}{\sqrt{{\sigma }^{2}T}}+\frac{18B^2}{T}+\frac{6\left(\bar{y}+y\right)B}{\sqrt{{\sigma }^2{T}^3}}\right),\\ {\mathrm{\Phi }}_{\bar{x}}& =\frac{12B}{\sqrt{{\sigma }^2{T}^3}},{\mathrm{\Phi }}_{\bar{y}}=\frac{A-6B}{\sqrt{{\sigma }^{2}T}},{\mathrm{\Phi }}_{\bar{y}\bar{y}}=\frac{4}{{\sigma }^{2}T},\end{array}$(3.32)

$\begin{array}{r}\frac{{\varpi }_{\bar{t}}}{\varpi }=-\frac{2}{T}-{\mathrm{\Phi }}_{\bar{t}},\frac{{\varpi }_{\bar{x}}}{\varpi }=-{\mathrm{\Phi }}_{\bar{x}},\frac{{\varpi }_{\bar{y}}}{\varpi }=-{\mathrm{\Phi }}_{\bar{y}},\frac{{\varpi }_{\bar{y}\bar{y}}}{\varpi }=-{\mathrm{\Phi }}_{\bar{y}\bar{y}}+{\mathrm{\Phi }}_{\bar{x}}^2,\end{array}$(3.33)

so that

$\begin{array}{rl}{\varpi }_{\bar{t}}^K-& \frac{1}{2}{\sigma }^2{\varpi }_{\bar{y}\bar{y}}^K+\bar{y}{\varpi }_{\bar{x}}^K+b{\varpi }_{\bar{y}}^K\\ =& {\varpi }^K\left(-\frac{2}{T}-{\mathrm{\Phi }}_{\bar{t}}+\frac{1}{2}{\sigma }^2\left({\mathrm{\Phi }}_{\bar{y}\bar{y}}-{\mathrm{\Phi }}_{\bar{y}}^2\right)-\bar{y}{\mathrm{\Phi }}_{\bar{x}}-b{\mathrm{\Phi }}_{\bar{y}}\right)\\ =& {\varpi }^K\left(-\frac{2}{T}+\frac{A^2}{2T}+\frac{bA}{\sqrt{{\sigma }^{2}T}}+\frac{18B^2}{T}+\frac{6\left(\bar{y}+y\right)B}{\sqrt{{\sigma }^2{T}^3}}\right.\\ & \left.+\frac{2}{T}-\frac{{\left(A-6B\right)}^2}{2T}-\frac{12\bar{y}B}{\sqrt{{\sigma }^2{T}^3}}-\frac{b\left(A-6B\right)}{\sqrt{{\sigma }^{2}T}}\right)=0.\end{array}$(3.34)

When $T\to 0,$ one has the following asymptotic expression:

$\begin{array}{r}{\varpi }^K\left(t,x,y,\bar{t},\bar{x},\bar{y}\right)\sim H\left(\frac{{\sigma }^2{T}^3}{12},\bar{x}-x\right)H\left({\sigma }^{2}T,\bar{y}-y\right)\to \delta \left(\bar{x}-x\right)\delta \left(\bar{y}-y\right).\end{array}$(3.35)

3.5 Solution via Kelvin Waves

Now, Kolmogorov’s formula is derived by using Kelvin waves (or an affine ansatz), which requires solving the problem of the following form:

$\begin{array}{rl}& {\mathcal{K}}_{\bar{t}}\left(t,x,y,\bar{t},\bar{x},\bar{y},k,l\right)-\frac{1}{2}{\sigma }^2{\mathcal{K}}_{\bar{y}\bar{y}}\left(t,x,y,\bar{t},\bar{x},\bar{y},k,l\right)\\ +& \bar{y}{\mathcal{K}}_{\bar{x}}\left(t,x,y,\bar{t},\bar{x},\bar{y},k,l\right)+b{\mathcal{K}}_{\bar{y}}\left(t,x,y,\bar{t},\bar{x},\bar{y},k,l\right)=0,\\ & \mathcal{K}\left(t,\bar{x},\bar{y},t,x,y,k,l\right)=exp\left(ik\left(\bar{x}-x\right)+il\left(\bar{y}-y\right)\right).\end{array}$(3.36)

Here

$\begin{array}{r}\left[k\right]=\frac{1}{L},\left[l\right]=\frac{T}{L},\left[\mathcal{K}\right]=1.\end{array}$(3.37)

By using the well-known results concerning the inverse Fourier transform of the $\delta$-function, one gets the following expression for the t.p.d.f. $\varpi \left(t,x,y,\bar{t},\bar{x},\bar{y}\right)$:

$\begin{array}{r}\varpi \left(t,x,y,\bar{t},\bar{x},\bar{y}\right)=\frac{1}{{\left(2\pi \right)}^2}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathcal{K}\left(t,x,y,\bar{t},\bar{x},\bar{y},k,l\right)dkdl.\end{array}$(3.38)

To calculate $\mathcal{K},$ one can use the affine ansatz and represent it in the following form:

$\begin{array}{r}\mathcal{K}\left(t,x,y,\bar{t},\bar{x},\bar{y},k,l\right)=exp\left(\mathrm{\Psi }\left(t,x,y,\bar{t},\bar{x},\bar{y},k,l\right)\right),\end{array}$(3.39)

where

$\begin{array}{r}\mathrm{\Psi }\left(t,x,y,\bar{t},\bar{x},\bar{y},k,l\right)=\alpha \left(t,\bar{t}\right)+ik\left(\bar{x}-x\right)+i\gamma \left(t,\bar{t}\right)\bar{y}-ily.\end{array}$(3.40)

and

$\begin{array}{rl}\frac{{\mathcal{K}}_{\bar{t}}}{\mathcal{K}}& ={\mathrm{\Psi }}_{\bar{t}}=\left({\alpha }_{\bar{t}}\left(t,\bar{t}\right)+i{\gamma }_{\bar{t}}\left(t,\bar{t}\right)\bar{y}\right),\frac{{\mathcal{K}}_{\bar{x}}}{\mathcal{K}}={\mathrm{\Psi }}_{\bar{x}}=ik,\\ \frac{{\mathcal{K}}_{\bar{y}}}{\mathcal{K}}& ={\mathrm{\Psi }}_{\bar{y}}=i\gamma \left(t,\bar{t}\right),\frac{{\mathcal{K}}_{\bar{y}\bar{y}}}{\mathcal{K}}={\mathrm{\Psi }}_{\bar{y}}^2=-{\gamma }^2\left(t,\bar{t}\right).\end{array}$(3.41)

Accordingly,

$\begin{array}{rl}{\alpha }_{\bar{t}}\left(t,\bar{t}\right)+\frac{1}{2}{\sigma }^2{\gamma }^2\left(t,\bar{t}\right)+i{\gamma }_{\bar{t}}\left(t,\bar{t}\right)\bar{y}+ik\bar{y}+ib\gamma \left(t,\bar{t}\right)& =0,\\ \alpha \left(t,t\right)=0,\gamma \left(t,t\right)& =l,\end{array}$(3.42)

so that

$\begin{array}{r}{\alpha }_{\bar{t}}\left(t,\bar{t}\right)+\frac{1}{2}{\sigma }^2{\gamma }^2\left(t,\bar{t}\right)+ib\gamma \left(t,\bar{t}\right)=0,\alpha \left(t,t\right)=0,\\ {\gamma }_{\bar{t}}\left(t,\bar{t}\right)+k=0,\gamma \left(t,t\right)=l.\end{array}$(3.43)

Straightforward calculation shows that:

$\begin{array}{rl}\gamma \left(t,\bar{t}\right)& =-kT+l,\\ \alpha \left(t,\bar{t}\right)& =-\frac{1}{2}{\sigma }^2\left(\frac{k^2{T}^3}{3}-kl{T}^2+l^{2}T\right)-ib\left(-\frac{k{T}^2}{2}+lT\right).\end{array}$(3.44)

Equations (3.38), (3.39), (3.40) and (3.44) yield

$\begin{array}{rl}& \varpi \left(t,x,y,\bar{t},\bar{x},\bar{y}\right)\\ & =\frac{1}{{\left(2\pi \right)}^2}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}exp\left(-\frac{1}{2}{\sigma }^2\left(\frac{k^2{T}^3}{3}-kl{T}^2+l^{2}T\right)\right.\\ & \left.+ik\left(\bar{x}-x-\bar{y}T+\frac{b{T}^2}{2}\right)+il\left(\bar{y}-y-bT\right)\right)dkdl.\end{array}$(3.45)

It is clear that $\varpi \left(t,x,y,\bar{t},\bar{x},\bar{y}\right)$ can be viewed as the characteristic function of the Gaussian density in the $\left(k,l\right)$ space, evaluated at the point $\left(\bar{x}-x-\bar{y}T\right.$ $\left.+\frac{b{T}^2}{2},\bar{y}-y-bT\right)$:

$\begin{array}{rl}\varpi & \left(t,x,y,\bar{t},\bar{x},\bar{y}\right)\\ & =\frac{{\left(det\left(\mathfrak{C}\right)\right)}^{1/2}}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}G\left(T,k,l\right)\\ & \times exp\left(ik\left(\bar{x}-x-\bar{y}T+\frac{b{T}^2}{2}\right)+il\left(\bar{y}-y-bT\right)\right)dkdl,\end{array}$(3.46)

where

$\begin{array}{r}G\left(T,k,l\right)=\frac{1}{2\pi {\left(det\left(\mathfrak{C}\right)\right)}^{1/2}}exp\left(-\frac{1}{2}\left(\begin{array}{c}k\\ l\end{array}\right)\cdot {\mathfrak{C}}^{-1}\left(T\right)\left(\begin{array}{c}k\\ l\end{array}\right)\right),\end{array}$(3.47)

and

$\begin{array}{rl}\mathfrak{C}\left(T\right)& =\left(\begin{array}{cc}\frac{12}{{\sigma }^2{T}^3}& \frac{6}{{\sigma }^2{T}^2}\\ \frac{6}{{\sigma }^2{T}^2}& \frac{4}{{\sigma }^{2}T}\end{array}\right),\\ \\ det\left(\mathfrak{C}\left(T\right)\right)& =\frac{12}{{\sigma }^4{T}^4}.\end{array}$(3.48)

As before, $\cdot$ denotes the scalar product. Accordingly,

$\begin{array}{r}\varpi \left(t,x,y,\bar{t},\bar{x},\bar{y}\right)=\frac{\sqrt{3}}{\pi {\sigma }^2{T}^2}exp\left(-\mathrm{\Omega }\left(t,x,y,\bar{t},\bar{x},\bar{y}\right)\right),\end{array}$(3.49)

where