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Darboux Transformation of Diffusion Processes

Alexey Kuznetsov, Minjian Yuan

TL;DR

This work develops a diffusion-theoretic incarnation of the Darboux transformation by combining Doob's $h$-transform, Siegmund duality, and a subsequent $h$-transform to produce a transformed diffusion $\tilde Y$ whose transition density is explicitly linked to the original $Y$ via a concise formula that depends on a positive seed $h$ solving ${\rm L}_Y h=\lambda h$. The main theoretical result provides a precise identity relating $p_t^{\tilde Y}$ to $p_t^Y$ through $h$ and $\,\lambda$, under suitable boundary and finiteness assumptions, thereby extending Siegmund duality to a broader class of boundary conditions. The paper then applies the construction to five Brownian-motion–based examples, deriving closed-form transition densities and, in four cases, spectral representations, and revealing how the Darboux transform shifts or augments the spectrum (e.g., adding a new eigenfunction $1/h$ or shifting eigenvalues by $-2\lambda$). These examples connect the probabilistic transform to Sturm–Liouville theory, Krein dual strings, and propagators for Schrödinger-type problems, highlighting potential applications in mathematical finance and physics. Overall, the results significantly broaden the set of diffusions with tractable transition densities by translating classical operator techniques into stochastic-analytic tools.

Abstract

Darboux transformation of a second-order linear differential operator is a well-known technique with many applications in mathematics and physics. We study Darboux transformation from the point of view of Markov semigroups of diffusion processes. We construct the Darboux transform of a diffusion process through a combination of Doob's $h$-transform and a version of Siegmund duality. Our main result is a simple formula that connects transition probability densities of the two processes. We provide several examples of Darboux transformed diffusion processes related to Brownian motion and Ornstein-Uhlenbeck process. For these examples, we compute explicitly the transition probability density and derive its spectral representation.

Darboux Transformation of Diffusion Processes

TL;DR

This work develops a diffusion-theoretic incarnation of the Darboux transformation by combining Doob's -transform, Siegmund duality, and a subsequent -transform to produce a transformed diffusion whose transition density is explicitly linked to the original via a concise formula that depends on a positive seed solving . The main theoretical result provides a precise identity relating to through and , under suitable boundary and finiteness assumptions, thereby extending Siegmund duality to a broader class of boundary conditions. The paper then applies the construction to five Brownian-motion–based examples, deriving closed-form transition densities and, in four cases, spectral representations, and revealing how the Darboux transform shifts or augments the spectrum (e.g., adding a new eigenfunction or shifting eigenvalues by ). These examples connect the probabilistic transform to Sturm–Liouville theory, Krein dual strings, and propagators for Schrödinger-type problems, highlighting potential applications in mathematical finance and physics. Overall, the results significantly broaden the set of diffusions with tractable transition densities by translating classical operator techniques into stochastic-analytic tools.

Abstract

Darboux transformation of a second-order linear differential operator is a well-known technique with many applications in mathematics and physics. We study Darboux transformation from the point of view of Markov semigroups of diffusion processes. We construct the Darboux transform of a diffusion process through a combination of Doob's -transform and a version of Siegmund duality. Our main result is a simple formula that connects transition probability densities of the two processes. We provide several examples of Darboux transformed diffusion processes related to Brownian motion and Ornstein-Uhlenbeck process. For these examples, we compute explicitly the transition probability density and derive its spectral representation.
Paper Structure (12 sections, 8 theorems, 88 equations, 1 table)

This paper contains 12 sections, 8 theorems, 88 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Siegmund duality
  4. Darboux transform of killed Brownian motion
  5. Examples
  6. Brownian motion on $\boldsymbol{{\mathbb R}}$
  7. Brownian motion on $\boldsymbol{(0,\infty)}$, killed at $\boldsymbol{0}$
  8. Brownian motion on $\boldsymbol{(0,\infty)}$, killed elastically at $\boldsymbol{0}$
  9. Brownian motion on $\boldsymbol{(0,1)}$, killed at $\boldsymbol{0}$ or $\boldsymbol{1}$
  10. Brownian motion on $\boldsymbol{{\mathbb R}}$ killed at rate $\boldsymbol{y^2/2}$
  11. Proof of identities (\ref{['p_BM_R_spectral']}) and (\ref{['tilde_p_spectral2']})
  12. Proof of (ii)

Key Result

Theorem 3.1

${}$ Assume that $(l,r) \subseteq {\mathbb R}$, $b(x) \in C((l,r))$, $\sigma(x)\in C^1((l,r))$ and $\sigma(x)>0$ for ${x \in (l,r)}$. Let $X$ be a conservative diffusion process on $(l,r)$ with infinitesimal generator and reflecting boundary conditions at every non-singular boundary point. Then there exists a diffusion process $\widetilde{X}$ such that for all $t>0$ and $x_i,y \in (l,r)$ for whi

Theorems & Definitions (12)

  • Theorem 3.1
  • Corollary 3.2
  • Remark 4.2
  • Proposition 4.4
  • Remark 4.5
  • Proposition 4.6
  • Definition 4.7
  • Theorem 4.8
  • Lemma 5.1
  • Corollary 5.2
  • ...and 2 more