Distributional Dynamics PDE

• Distributional Dynamics PDE is a class of partial differential equations that evolve functions or probability distributions using operators defined in a distributional sense, crucial for modeling complex systems.
• The framework relies on mollifier approximations, Sobolev/Besov space theory, and fixed point or monotonicity techniques to ensure existence and uniqueness of generalized solutions.
• A strong link with probability is established via associated Markov processes, SDEs, and BSDEs, culminating in a nonlinear Feynman-Kac formula that connects analytic and stochastic representations.

Distributional dynamics PDE refers to a broad class of partial differential equations where the evolution of a quantity—typically a function or a probability distribution—is governed by operators or nonlinearities that act in a “distributional sense.” This includes cases where coefficients, drift terms, or nonlinearities are generalized functions (distributions), as well as PDEs that describe the dynamics of probability distributions themselves (such as McKean–Vlasov equations, nonlinear Fokker–Planck equations, and mean-field models). The mathematical treatment and physical interpretation of such equations involve advanced tools from functional analysis, stochastic analysis, Sobolev and Besov space theory, and the theory of weak or generalized solutions. These frameworks are critical for modeling complex systems with singularities, random environments, mean-field interactions, and for connecting analytic and probabilistic representations.

1. Semilinear PDEs with Distributional Drift

A prototypical example is a semilinear elliptic PDE where the drift coefficient is only defined in the sense of distributions. The general form considered is: $$L g(x) = \frac{\sigma^2(x)}{2} g''(x) + \beta'(x) g'(x)$$ where $\sigma > 0$ is a continuous function, and $\beta \in C^0$ with distributional derivative $\beta'$. To make sense of $L$, one introduces mollified approximations: $$\sigma_n^2 = \sigma^2 * \Phi_n, \quad \beta_n = \beta * \Phi_n, \quad L_n g = \frac{\sigma_n^2}{2} g'' + \beta_n' g'$$ A function $u \in C^1$ is a generalized $C^1$ solution to $Lu = F(x, u, u')$ if there exist sequences $f_n$, $\ell_n$ solving $L_n f_n = \ell_n$ and converging to $u$ and $F(x,u,u')$, respectively.

The explicit representation of the derivative of $u$ is given by: $$u'(x) = e^{-\Sigma(x)}\Biggl(2 \int_a^x \frac{e^{\Sigma(y)} F(y,u(y),u'(y))}{\sigma^2(y)} \, dy + u'(a)\Biggr)$$ with $$\Sigma(x) := \lim_{n \to \infty} 2 \int_a^x \frac{\beta_n'(y)}{\sigma_n^2(y)} dy$$.

This generalized framework admits solutions even when $\beta'$ is highly singular, such as in models of diffusion with random environments or Brox diffusion (Russo et al., 2014).

2. Existence, Uniqueness, and Methods of Solution

Existence and uniqueness of generalized solutions depend on Lipschitz and monotonicity conditions on the nonlinearity $F$. Existence is typically established via:

• Contraction mapping / fixed point arguments: Using the integral equation formulation with an appropriate kernel, a contraction property is shown under small Lipschitz constant.
• Monotonicity arguments: If $F$ is monotone in $u$, uniqueness and comparison principles are established, notably that if $$(F(x, y, z) - F(x, \tilde{y}, z))(y - \tilde{y}) \geq 0$$ for all $x$, $y$, $\tilde{y}$, $z$, then two $C^1$-solutions must coincide.

These allow the paper of boundary value problems for such singular PDEs and underpin the well-posedness theory for elliptic equations with distributional drift (Russo et al., 2014).

3. Probabilistic Correspondence: Markov Processes and Martingale Problem

The operator $L$ can be regarded as the generator of a Markov process $X$ that solves a stochastic differential equation (SDE) with a distributional drift: $dX_t = \beta'(X_t)dt + \sigma(X_t) dW_t$ Since $\beta'$ is only a distribution, $X$ is defined as a weak solution via the martingale problem: $$f(X_t) - f(x_0) - \int_0^t Lf(X_s)ds \quad \text{is a local martingale}$$ for all smooth test functions $f$. Under suitable regularity and monotonicity, uniqueness in law for $X$ is achieved (Russo et al., 2014).

This probabilistic view links the analytic definition of $L$ and its PDEs to stochastic processes, enabling representation theorems and probabilistic proofs of existence and uniqueness.

4. Backward SDEs with Distributional Drivers and the Nonlinear Feynman–Kac Formula

There is a deep connection between solutions of distributional dynamics PDEs and backward stochastic differential equations (BSDEs) with potentially singular drivers. If $u$ is a solution of the semilinear PDE, one constructs the BSDE solution along the path of $X$ by

$$Y_t = u(X_{t \wedge \tau}), \quad Z_t = u'(X_t)\mathbf{1}_{[0, \tau]}(t), \quad O \equiv 0$$

where $\tau$ is the first exit time from the interval. This triple $(Y, Z, O)$ solves the BSDE

$$Y_t = \xi - \int_t^\infty Z_s dM_s + \int_t^\infty f(\omega, s, Y_s, Z_s)d\langle M\rangle_s - (O_\tau - O_{t \wedge \tau})$$

with driver $$f(\omega, t, y, z) = -\frac{F(X_t(\omega), y, z)}{\sigma^2(X_t(\omega))}$$ and $\xi = u(X_\tau)$.

Such representation establishes the so-called nonlinear Feynman-Kac formula for equations with distributional coefficients and allows one to treat random terminal times driven by general cadlag martingales, not necessarily Brownian motion (Russo et al., 2014).

5. Analytic and Probabilistic Techniques for Distributional Dynamics PDEs

Working with distributional coefficients requires advanced analytic tools:

• Mollifier approximations: All singular coefficients are regularized using convolution with smooth mollifiers, with results passed to the limit.
• Integral representation formulas: Explicit integral expressions for the solution and its derivatives permit fixed-point or monotonicity analysis even in the presence of irregular coefficients.
• Functional analytic spaces: Solutions and coefficients are often handled in Sobolev, Besov, or Hölder-type spaces with negative regularity, ensuring that products (such as $F(Vu) \cdot b$) are rigorously defined.
• Martingale and stopping time techniques: On the probabilistic side, solutions are constructed using martingale and stopping time arguments leveraging the weak formulation of SDEs.

These methods collectively enable rigorous treatment of distributional dynamics PDEs, even when classical solution concepts are unavailable.

6. Interplay and Broader Impact

The integration of analytic and probabilistic methods for distributional dynamics PDEs has advanced the mathematical understanding of models arising in disordered systems, random media, kinetic theory, and stochastic control. By establishing a bridge with BSDE theory—where the distributional drift in the PDE translates into the driver or the law of the forward process—one obtains new representation results and uniqueness principles even in weak or generalized solution settings.

The approach is essential for extending classical SDE/PDE theories to settings where the drift is merely a distribution (e.g., in homogenization limits, turbulence modeling, and kinetic equations with measure-valued data). The fusion of stochastic analysis, limiting procedures via mollifiers, and variational methods in functional analytic spaces represents a defining feature of modern distributional dynamics PDE theory.

This framework continues to motivate new probabilistic and analytic techniques for highly irregular, nonlinear, and distribution-dependent evolution equations, and enables applications ranging from disordered physical systems to mathematical finance and stochastic numerical analysis.