Deriving Reverse-Time Stochastic Differential Equations (SDEs)

Introduction

The reverse-time formulation of a diffusion process is central to many applications, including score-based generative modeling. While the forward process is governed by a stochastic differential equation (SDE), the corresponding reverse-time process also follows an SDE, albeit with a modified drift term. Deriving this reverse SDE requires careful handling of time reversal and the associated Fokker-Planck equations.

In their seminal work on score-based generative modeling, Song et al., 2021 utilize the reverse-time SDE but refer to an earlier source (Anderson, 1982) for its derivation. While the result presented there is correct under common assumptions (like time-dependent diffusion coefficients), the general derivation involves an additional term. This post provides a self-contained and precise derivation for the general case in modern MathJax notation, clarifying the origin of all terms and following specific formatting guidelines.

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Main Derivation: Reverse-Time SDE

We consider a forward diffusion process \(X_t \in \mathbb{R}^d\) on the time interval \([0,T]\), given by the Itô SDE:

\[dX_t \;=\; f(X_t,t)\,dt \;+\; g(X_t,t)\,dW_t, \quad X_0 \text{ given},\]

where:

1. \(W_t\) is a standard \(d\)-dimensional Wiener process (Brownian motion) adapted to the forward filtration \(\{\mathcal{F}_t\}_{t\in[0,T]}\).
2. \(f: \mathbb{R}^d \times [0,T] \to \mathbb{R}^d\) is the drift coefficient.
3. \(g: \mathbb{R}^d \times [0,T] \to \mathbb{R}^{d \times d}\) is the diffusion coefficient matrix. We assume \(g(x,t)\) is such that \(g(x,t) g(x,t)^T\) is positive definite (uniform ellipticity). (For simplicity in notation, we will sometimes write \(g^2\) to mean \(g g^T\).)
4. Suitable conditions (e.g., global Lipschitz, linear growth) are assumed on \(f\) and \(g\) to ensure the existence and uniqueness of strong solutions.
5. \(p(x,t)\) denotes the probability density function of \(X_t\), assumed to be sufficiently smooth (e.g., \(\mathcal{C}^{2,1}\)) and strictly positive for \(t>0\).

Under these assumptions, the density \(p(x,t)\) satisfies the Fokker–Planck equation (also known as the forward Kolmogorov equation). See Appendix A.1 for a sketch of its derivation. Assuming \(g\) is scalar for simpler notation (i.e., \(g(x,t) \in \mathbb{R}\) and \(W_t\) is 1D, or \(g(x,t)\) is diagonal \(g(x,t)I\)), the Fokker-Planck equation is:

\[\frac{\partial p}{\partial t}(x,t) \;=\; -\nabla \cdot \bigl( f(x,t)\,p(x,t)\bigr) \;+\; \tfrac12 \,\Delta\!\Bigl( g(x,t)^2\,p(x,t)\Bigr).\]

(Note: For matrix-valued \(g\), the Laplacian term becomes \(\sum_{i,j} \frac{\partial^2}{\partial x_i \partial x_j} \left( \frac12 (g g^T)_{ij} p \right)\). We stick to the scalar/diagonal case notationally, but the logic extends.)

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1. Introducing Reverse Time

Define the reverse-time variable: \(s \;=\; T - t\), which means \(t \;=\; T - s\). As \(t\) goes from \(0\) to \(T\), \(s\) goes from \(T\) to \(0\). We consider the reverse-time process:

\[Y_s \;:=\; X_{T-s}, \quad s \in [0,T].\]

To simplify notation in reverse time, let:

\[F(x,s) \;:=\; f\bigl(x,\;T-s\bigr), \quad G(x,s) \;:=\; g\bigl(x,\;T-s\bigr), \quad q(x,s) \;:=\; p\bigl(x,\;T-s\bigr).\]

Note that \(q(x,s)\) is the probability density of \(Y_s\).

Now, we rewrite the forward Fokker–Planck equation in terms of \((x,s)\). Using the chain rule, \(\frac{\partial}{\partial t} = \frac{\partial s}{\partial t} \frac{\partial}{\partial s} = - \frac{\partial}{\partial s}\). Substituting this and the definitions of \(F, G, q\) into the Fokker-Planck equation gives:

\[-\frac{\partial q}{\partial s}(x,s) \;=\; -\nabla \cdot \Bigl(F(x,s)\,q(x,s)\Bigr) \;+\; \tfrac12\,\Delta\!\Bigl(G(x,s)^2\,q(x,s)\Bigr).\]

Multiplying by \(-1\), we get the evolution equation for the density \(q\) in reverse time \(s\):

\[\frac{\partial q}{\partial s}(x,s) \;=\; \nabla \cdot \Bigl(F(x,s)\,q(x,s)\Bigr) \;-\; \tfrac12\,\Delta\!\Bigl(G(x,s)^2\,q(x,s)\Bigr).\]

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2. Postulating an SDE for the Reverse Process

We postulate that the reverse process \(Y_s\) also satisfies an Itô SDE, but driven by a different Wiener process \(\widetilde{W}_s\) adapted to a reversed filtration \(\{\mathcal{G}_s\}_{s\in[0,T]}\) (where \(\mathcal{G}_s\) roughly contains information up to time \(T-s\) in the forward process). The postulated SDE form is:

\[dY_s \;=\; b(Y_s,s)\,ds \;+\; G(Y_s,s)\,d\widetilde{W}_s.\]

Here, \(b(x,s)\) is the unknown reverse drift coefficient we need to determine. Note that the diffusion coefficient \(G(x,s)\) is the same as in the forward SDE (evaluated at the corresponding time). This is a standard result in time reversal of diffusions (see Appendix A.2).

The Fokker–Planck equation associated with this postulated reverse SDE, governing the evolution of its density \(q(x,s)\), is:

\[\frac{\partial q}{\partial s}(x,s) \;=\; -\nabla\cdot\Bigl(b(x,s)\,q(x,s)\Bigr) \;+\; \tfrac12\,\Delta\!\Bigl(G(x,s)^2\,q(x,s)\Bigr).\]

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3. Matching the Density Evolutions

For the postulated SDE to be consistent with the time-reversed forward process, the density \(q(x,s)\) must satisfy both derived evolution equations. Therefore, we set the right-hand sides equal:

\[-\nabla \cdot \Bigl(b(x,s)\,q(x,s)\Bigr) \;+\; \tfrac12\,\Delta\!\Bigl(G(x,s)^2\,q(x,s)\Bigr) \;=\; \nabla \cdot \Bigl(F(x,s)\,q(x,s)\Bigr) \;-\; \tfrac12\,\Delta\!\Bigl(G(x,s)^2\,q(x,s)\Bigr).\]

Rearranging terms to isolate the drift terms involving \(b\) and \(F\):

\[-\nabla \cdot \Bigl(b(x,s)\,q(x,s)\Bigr) \;-\; \nabla \cdot \Bigl(F(x,s)\,q(x,s)\Bigr) \;=\; -\Delta\!\Bigl(G(x,s)^2\,q(x,s)\Bigr).\]

Combine the divergence terms and multiply by \(-1\):

\[\nabla \cdot \Bigl( [b(x,s) + F(x,s)] \, q(x,s) \Bigr) \;=\; \Delta\!\Bigl(G(x,s)^2\,q(x,s)\Bigr).\]

Recall that \(\Delta = \nabla \cdot \nabla\). So, the equation is:

\[\nabla \cdot \Bigl( [b(x,s) + F(x,s)] \, q(x,s) \Bigr) \;=\; \nabla \cdot \Bigl[ \nabla \bigl( G(x,s)^2\,q(x,s) \bigr) \Bigr].\]

This equation is a statement in terms of distributions. To extract a relationship between the coefficients, we can integrate against a smooth test function \(\phi \in \mathcal{C}^\infty_c(\mathbb{R}^d)\) and use integration by parts (divergence theorem). For any such \(\phi\):

\[\int \nabla \cdot \Bigl( [b + F] \, q \Bigr) \phi \, dx \;=\; \int \nabla \cdot \Bigl[ \nabla \bigl( G^2\,q \bigr) \Bigr] \phi \, dx\]

Applying integration by parts to both sides (boundary terms vanish because \(\phi\) has compact support):

\[- \int \Bigl( [b(x,s) + F(x,s)] \, q(x,s) \Bigr) \cdot \nabla \phi(x) \, dx \;=\; - \int \Bigl[ \nabla \bigl( G(x,s)^2\,q(x,s) \bigr) \Bigr] \cdot \nabla \phi(x) \, dx\]

Since this equality must hold for all test functions \(\phi\), the vector fields multiplying \(\nabla \phi(x)\) must be equal (almost everywhere):

\[[b(x,s) + F(x,s)] \, q(x,s) \;=\; \nabla \bigl( G(x,s)^2\,q(x,s) \bigr).\]

Now, we expand the gradient on the right-hand side using the product rule \(\nabla(uv) = (\nabla u)v + u(\nabla v)\):

\[\nabla \bigl( G(x,s)^2\,q(x,s) \bigr) \;=\; \Bigl[ \nabla G(x,s)^2 \Bigr] q(x,s) \;+\; G(x,s)^2 \nabla q(x,s).\]

Here, \(\nabla G(x,s)^2\) denotes the gradient with respect to \(x\) of the function \(G(x,s)^2\).

Substituting this back yields:

\[[b(x,s) + F(x,s)] \, q(x,s) \;=\; \Bigl[ \nabla G(x,s)^2 \Bigr] q(x,s) \;+\; G(x,s)^2 \nabla q(x,s).\]

We are looking for \(b(x,s)\). Rearranging the terms:

\[b(x,s) \, q(x,s) \;=\; -\,F(x,s) \, q(x,s) \;+\; \Bigl[ \nabla G(x,s)^2 \Bigr] q(x,s) \;+\; G(x,s)^2 \nabla q(x,s).\]

Since we assumed the density \(q(x,s) = p(x, T-s)\) is strictly positive, we can divide both sides by \(q(x,s)\):

\[b(x,s) \;=\; -\,F(x,s) \;+\; \nabla G(x,s)^2 \;+\; G(x,s)^2 \,\frac{\nabla q(x,s)}{q(x,s)}.\]

Recognizing that \(\frac{\nabla q(x,s)}{q(x,s)} = \nabla \log q(x,s)\) (the score function of the density \(q\)), we obtain the final expression for the reverse drift:

\[b(x,s) \;=\; -\,F(x,s) \;+\; \nabla G(x,s)^2 \;+\; G(x,s)^2\,\nabla \log q(x,s).\]

Important Special Case: In many applications, including standard setups for diffusion models, the diffusion coefficient \(g\) in the forward SDE depends only on time, i.e., \(g(t)\). In this case, \(G(x,s) = g(T-s)\) also depends only on time (\(s\)), not on space (\(x\)). Consequently, its spatial gradient is zero: \(\nabla G(x,s)^2 = \nabla_x G(s)^2 = \mathbf{0}\). Under this simplifying assumption, the reverse drift becomes:

\[b(x,s) \;=\; -\,F(x,s) \;+\; G(s)^2\,\nabla \log q(x,s).\]

This is the form often cited in literature focused on these specific models. Our derivation shows the more general form includes the \(\nabla G^2\) term when \(g\) depends on space.

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4. Final Form of the Reverse-Time SDE

Substituting the derived drift \(b(x,s)\) back into the postulated reverse SDE, we get the general reverse-time SDE for \(Y_s = X_{T-s}\):

\[dY_s \;=\; \Bigl[-\,F(Y_s,s) \;+\; \nabla G(Y_s,s)^2 \;+\; G(Y_s,s)^2\,\nabla \log q(Y_s,s)\Bigr]\,ds \;+\; G(Y_s,s)\,d\widetilde{W}_s.\]

Expressing this entirely in terms of the original forward process functions \(f, g\) and the forward density \(p\):

\[dY_s \;=\; \Biggl[-\,f\bigl(Y_s,\;T-s\bigr) \;+\; \nabla_x\Bigl[g\bigl(Y_s,\;T-s\bigr)^2\Bigr] \;+\; g\bigl(Y_s,\;T-s\bigr)^2\,\nabla_x \log p\bigl(Y_s,\;T-s\bigr)\Biggr]\,ds \;+\; g\bigl(Y_s,\;T-s\bigr)\,d\widetilde{W}_s.\]

(Here, \(\nabla_x\) emphasizes the gradient is taken with respect to the spatial variable, which is the argument of the functions.)

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Appendix

A.1. Sketch of the Fokker–Planck Derivation

For the forward SDE

\[dX_t \;=\; f(X_t,t)\,dt \;+\; g(X_t,t)\,dW_t,\]

let \(\phi\in C_c^\infty(\mathbb{R}^d)\) be a smooth test function with compact support. By Itô’s formula (using scalar \(g\) for simplicity):

\[d\,\phi(X_t) \;=\; \Biggl[f(X_t,t)\cdot\nabla\phi(X_t) \;+\; \tfrac12\,g(X_t,t)^2\,\Delta\phi(X_t)\Biggr]\,dt \;+\; g(X_t,t)\,\nabla\phi(X_t)\,dW_t.\]

Taking expectations, noting that the expectation of the Itô integral term (which is a martingale under suitable conditions) is zero:

\[\frac{d}{dt}\,\mathbb{E}[\phi(X_t)] \;=\; \mathbb{E}\Biggl[f(X_t,t)\cdot\nabla\phi(X_t) \;+\; \tfrac12\,g(X_t,t)^2\,\Delta\phi(X_t)\Biggr].\]

Expressing the expectations using the probability density \(p(x,t)\) of \(X_t\):

\[\mathbb{E}[\phi(X_t)] \;=\; \int \phi(x)\,p(x,t)\,dx,\]

so

\[\frac{d}{dt}\,\mathbb{E}[\phi(X_t)] \;=\; \int \phi(x)\,\frac{\partial p}{\partial t}(x,t)\,dx.\]

The right-hand side becomes:

\[\mathbb{E}[\dots] \;=\; \int \Biggl[f(x,t)\cdot\nabla\phi(x) \;+\; \tfrac12\,g(x,t)^2\,\Delta\phi(x)\Biggr]\, p(x,t)\,dx.\]

Equating the two expressions for the time derivative:

\[\int \phi(x)\,\frac{\partial p}{\partial t}(x,t)\,dx \;=\; \int \Biggl[f(x,t)\cdot\nabla\phi(x) \;+\; \tfrac12\,g(x,t)^2\,\Delta\phi(x)\Biggr]\, p(x,t)\,dx.\]

Now, perform integration by parts on the right-hand side terms to move derivatives from \(\phi\) to the other factors. The boundary terms vanish due to the compact support of \(\phi\).

1. \[\int (f \cdot \nabla \phi) p \,dx = - \int \phi \, \nabla \cdot (f p) \,dx\]
2. \[\int (\tfrac12 g^2 \Delta \phi) p \,dx = \int (\tfrac12 g^2 \nabla \cdot (\nabla \phi)) p \,dx = - \int \nabla \phi \cdot \nabla(\tfrac12 g^2 p) \,dx = \int \phi \, \Delta(\tfrac12 g^2 p) \,dx\]

Substituting these back:

\[\int \phi(x)\,\frac{\partial p}{\partial t}(x,t)\,dx \;=\; \int \phi(x) \Biggl[ -\nabla \cdot \bigl(f(x,t)\,p(x,t)\bigr) \;+\; \tfrac12\,\Delta\!\Bigl(g(x,t)^2\,p(x,t)\Bigr) \Biggr] \, dx.\]

Since this identity holds for all test functions \(\phi\), the integrands must be equal, yielding the Fokker–Planck equation:

\[\frac{\partial p}{\partial t}(x,t) \;=\; -\nabla\cdot\bigl(f(x,t)\,p(x,t)\bigr) \;+\; \tfrac12\,\Delta\!\Bigl(g(x,t)^2\,p(x,t)\Bigr).\]

A.2. Why the Reverse Process Follows an SDE

The claim that the time-reversed process \(Y_s = X_{T-s}\) satisfies an SDE of the Itô form relies on deep results from stochastic calculus concerning time reversal of Markov processes. Here’s a brief outline:

A.2.1. Markov Property and Martingale Representation

• Markov Property Preservation: If the forward process \(X_t\) is a Markov process (which solutions to SDEs typically are under sufficient regularity), then under suitable conditions (smoothness, non-degeneracy/ellipticity of \(g g^T\)), the reversed process \(Y_s\) is also a Markov process with respect to the reverse filtration \(\mathcal{G}_s = \sigma(Y_u : u \ge s)\) (or more formally defined via the forward filtration). Proving this rigorously involves showing conditional independence properties using tools like Bayes’ theorem on densities or more abstract measure-theoretic arguments.
• Martingale Representation Theorem: Since \(Y_s\) is a Markov process adapted to the filtration \(\{\mathcal{G}_s\}\), the Martingale Representation Theorem applies. It implies that any \(\mathcal{G}_s\)-martingale can be represented as a stochastic integral with respect to some \(\mathcal{G}_s\)-Brownian motion, which we denoted \(\widetilde{W}_s\). The existence of such a representation theorem is key to asserting that the “random part” of \(dY_s\) can be captured by a \(d\widetilde{W}_s\) term. The specific form \(G(Y_s,s) d\widetilde{W}_s\) arises from the structure of the quadratic variation of the process, which is preserved under time reversal.

A.2.2. Construction of the Backward Wiener Process

• Backward Filtration: The natural filtration for the reverse process is \(\mathcal{G}_s = \sigma(X_u : u \le T-s)\), potentially augmented. This is a decreasing family of \(\sigma\)-algebras (\(\mathcal{G}_s \supseteq \mathcal{G}_t\) if \(s < t\)). A process \(\widetilde{W}_s\) is a Brownian motion with respect to \(\{\mathcal{G}_s\}\) if it is \(\mathcal{G}_s\)-adapted, has independent increments into the past (i.e., \(\widetilde{W}_t - \widetilde{W}_s\) is independent of \(\mathcal{G}_t\) for \(s < t\)), and increments have the correct Gaussian distribution. Constructing such a process is non-trivial but possible under appropriate conditions.
• Backward Itô Integral: Stochastic integrals with respect to \(\widetilde{W}_s\) can be defined (e.g., the backward Itô integral). The reverse-time SDE is interpreted in this framework. The consistency requirement between the SDE dynamics and the evolution of the density \(q(x,s)\) (via the Fokker-Planck equation) then uniquely determines the drift term \(b(x,s)\).

A.2.3. References

For rigorous proofs and detailed discussion on the time reversal of diffusion processes:

• Anderson, B. D. O. (1982). Reverse-time diffusion equation models. Stochastic Processes and their Applications, 12(3), 313-326. (While the original reference, it’s concise; book treatments might be clearer).
• Haussmann, U. G., & Pardoux, E. (1986). Time reversal of diffusions. The Annals of Probability, 14(4), 1188–1205. (A key theoretical paper).
• Øksendal, B. (2003). Stochastic Differential Equations: An Introduction with Applications. Springer. (Chapter 11 in some editions touches upon time reversal).

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Final Summary

1. Forward SDE & Fokker–Planck: The forward process \(X_t\) solves \(dX_t = f(X_t,t)\,dt + g(X_t,t)\,dW_t\), with density \(p(x,t)\) satisfying the Fokker-Planck equation.
2. Reverse-Time Definition: The reverse process is \(Y_s = X_{T-s}\). We define \(F(x,s) = f(x,T-s)\), \(G(x,s) = g(x,T-s)\), and the reverse density \(q(x,s) = p(x,T-s)\).
3. Reverse SDE Postulate: We assume \(dY_s = b(Y_s,s)\,ds + G(Y_s,s)\,d\widetilde{W}_s\).
4. Matching Densities: By requiring the Fokker-Planck equation for the reverse SDE to match the time-transformed forward Fokker-Planck equation, we derive the reverse drift \(b(x,s)\).

5. Reverse Drift Result: The general reverse drift is:

   \[b(x,s) = -F(x,s) + \nabla G(x,s)^2 + G(x,s)^2\,\nabla \log q(x,s).\]
6. Special Case: If \(g\) (and thus \(G\)) depends only on time, \(\nabla G^2 = \mathbf{0}\), simplifying the drift to \(b(x,s) = -F(x,s) + G(s)^2\,\nabla \log q(x,s)\).
7. Reverse SDE: The final reverse-time SDE is obtained by substituting the derived \(b(x,s)\) into the postulated SDE form.
8. Theoretical Basis (Appendix A.2): The existence of such a reverse SDE relies on the preservation of the Markov property under time reversal and the Martingale Representation Theorem applied to the reversed filtration.