docs(v2.0.0-alpha.5): rich math+physics background per chapter, fix notebook download links

Each of the eight v2.0 algorithm pages (bsde, pde, stochastic_control,
quadratic_impact_control, mckean_vlasov, agent_based, robust_drift,
generative_calibration_hooks) gains:

- A dedicated 'Mathematical background' section with the central theorem
  (Pardoux-Peng, Sznitman propagation of chaos, Pontryagin-Bismut,
  Huber-IRLS, Gretton MMD, Kolmogorov forward, etc.), key derivations
  and the analytic closed-form solution that the unit tests target.
- An 'Applications' / 'Why it matters' paragraph listing concrete
  research and engineering use-cases so newcomers grasp the value of
  each primitive.
- A repaired companion-notebook block: the broken relative path
  '../../examples/notebooks/...ipynb' (which 404s on RTD) is replaced
  by an explicit GitHub blob (view) + raw (download) URL pair.

Sphinx now builds the full doc set with zero new warnings.

docs/source/algorithms/pde.rst

@@ -1,9 +1,83 @@
 PDE — Fokker–Planck, HJB, elliptic Poisson
 ==========================================
 
-Three CPU-only finite-difference solvers: 1-D forward Fokker–Planck (`fokker_planck_constant`), 2-D explicit HJB (`hjb_quadratic_2d`) and 2-D Poisson SOR (`poisson_2d_zero_boundary`).  Each routine is verified against an analytic ground truth.
+Three CPU-only finite-difference solvers covering the two canonical PDE pillars of stochastic
+analysis: the **forward** equation for the marginal density of a diffusion (Fokker–Planck),
+the **backward** equation for an optimally controlled diffusion (Hamilton–Jacobi–Bellman),
+and a static **elliptic** boundary-value problem (Poisson).
 
-.. note:: Companion executed notebook: `11_pde.ipynb <../../examples/notebooks/11_pde.ipynb>`_
+Mathematical background
+-----------------------
+
+**Fokker–Planck (Kolmogorov forward).**  For a 1-D Itô diffusion
+$dX_t = \mu(t, x)\, dt + \sigma(t, x)\, dW_t$, the marginal density $\rho(t, x)$ of $X_t$
+satisfies the parabolic PDE
+
+.. math::
+
+   \partial_t \rho \;+\; \partial_x\!\bigl(\mu(t,x)\, \rho\bigr)
+   \;=\; \tfrac12\, \partial^2_{xx}\!\bigl(\sigma^2(t,x)\, \rho\bigr),
+   \qquad \rho(0, \cdot) = \rho_0 .
+
+For the *pure-diffusion* test ($\mu \equiv 0$, $\sigma^2 \equiv 1$, $\rho_0 = \mathcal{N}(0, 1)$)
+the analytic Gaussian heat kernel gives $\rho(t, x) = \frac{1}{\sqrt{2\pi(1+t)}}\exp\!\bigl(-\frac{x^2}{2(1+t)}\bigr)$,
+so the variance grows linearly: $\mathrm{Var}(X_t) = 1 + t$.  The conservative
+Lax–Wendroff / centred-flux scheme implemented by `fokker_planck_constant` preserves total mass
+(checked in the notebook to machine precision).
+
+**Hamilton–Jacobi–Bellman.**  Consider the controlled diffusion
+$dX_t = \mu(X_t, \alpha_t)\, dt + \sigma(X_t)\, dW_t$ and the value function
+$v(t, x) = \sup_\alpha \mathbb{E}_{t,x}\!\bigl[\int_t^T r(X_s, \alpha_s)\, ds + g(X_T)\bigr]$.
+Dynamic programming produces
+
+.. math::
+
+   \partial_t v \;+\; \sup_{a \in \mathcal{A}}\Bigl\{ \mu(x, a) \cdot \nabla v
+     \;+\; \tfrac12\, \mathrm{tr}\!\bigl(\sigma\sigma^\top(x)\, \nabla^2 v\bigr)
+     \;+\; r(x, a) \Bigr\} \;=\; 0,
+   \qquad v(T, x) = g(x).
+
+`hjb_quadratic_2d` discretises this in 2-D by an explicit finite-difference scheme; the simple
+heat-only relaxation case (:math:`H \equiv 0`, :math:`\sigma^2 > 0`) preserves a constant value while a
+quadratic terminal :math:`g(x) = \tfrac12 \lVert x \rVert^2` smooths into a Gaussian-shaped value surface.
+
+**Elliptic Poisson with zero Dirichlet boundary.**  On the unit square $\Omega = (0,1)^2$,
+
+.. math::
+
+   -\Delta u(x, y) = f(x, y) \text{ in } \Omega, \qquad u\!\restriction_{\partial\Omega} = 0 .
+
+The Laplace eigenfunctions $\phi_{m,n}(x, y) = \sin(m\pi x)\sin(n\pi y)$ form an
+orthonormal basis with eigenvalues $\lambda_{m,n} = (m^2 + n^2)\pi^2$, so for
+$f = 2\pi^2 \sin(\pi x)\sin(\pi y)$ the *exact* solution is
+$u(x, y) = \sin(\pi x)\sin(\pi y)$.  `poisson_2d_zero_boundary` solves the 5-point stencil by
+**Successive Over-Relaxation** with optimal relaxation parameter
+$\omega^* = 2 / (1 + \sin(\pi h))$ for grid spacing $h = 1/(N-1)$, achieving spectral radius
+$\rho \sim 1 - 2\pi h$ — i.e. $O(h^{-1})$ iterations to reach a fixed tolerance, against
+$O(h^{-2})$ for plain Gauss–Seidel.
+
+**Probabilistic representation (Feynman–Kac).**  Both the parabolic HJB and the elliptic
+Poisson PDE admit stochastic representations: $u(x) = \mathbb{E}_x\!\bigl[\int_0^{\tau_\Omega} f(X_s)\, ds\bigr]$
+for the latter, where $\tau_\Omega$ is the first exit time of the diffusion from $\Omega$.
+This links the PDE solvers above to the BSDE primitives of :doc:`bsde`.
+
+Why it matters
+--------------
+
+* **Density estimation under controlled noise.**  Fokker–Planck is the workhorse of
+  non-equilibrium statistical physics, plasma transport, calibration of stochastic-volatility
+  models, and Langevin-based MCMC convergence diagnostics.
+* **Optimal control & inverse problems.**  HJB is the cornerstone of dynamic programming,
+  reinforcement learning (continuous-time policy iteration), and stochastic-control routing.
+* **Mean-field games.**  The MFG fixed point is exactly the coupled system
+  *(backward HJB + forward Fokker–Planck)* with cost depending on the density — building this
+  loop on top of the two solvers above is one of the v2.0 milestones.
+* **Image processing & PDE-constrained optimisation.**  Poisson editing, electric-potential
+  reconstruction, gravitational-potential inversion all reduce to the same elliptic stencil.
+
+.. note::
+   📓 **Companion notebook** — `view on GitHub <https://github.com/ThotDjehuty/optimiz-rs/blob/main/examples/notebooks/11_pde.ipynb>`_
+   · `download .ipynb <https://raw.githubusercontent.com/ThotDjehuty/optimiz-rs/main/examples/notebooks/11_pde.ipynb>`_
 
 11 — PDE solvers
 ================