[Paper Review] Hörmander's theorem for stochastic partial differential equations

Н. В. Крылов|arXiv (Cornell University)|Sep 22, 2013

Stochastic processes and financial applications9 references3 citations

TL;DR

This paper establishes a local hypoellipticity result for stochastic partial differential equations (SPDEs) under a time-dependent Hörmander condition, proving that solutions are infinitely differentiable in space when the coefficients are only measurable in time. The method uses Wentzell's random change of coordinates to eliminate stochastic terms locally, reducing the SPDE to a deterministic parabolic equation with random coefficients, enabling application of classical regularity theory to achieve infinite differentiability under local Lie algebra rank conditions.

ABSTRACT

We prove Hörmander's type hypoellipticity theorem for stochastic partial differential equations when the coefficients are only measurable with respect to the time variable. The need for such kind of results comes from filtering theory of partially observable diffusion processes, when even if the initial system is autonomous, the observation process enters the coefficients of the filtering equation and makes them time-dependent with no good control on the smoothness of the coefficients with respect to the time variable.

Motivation & Objective

• To establish local hypoellipticity for SPDEs when coefficients are only measurable in time, a setting arising naturally in filtering theory of partially observed diffusions.
• To overcome limitations in prior works that required global conditions or Hölder continuity in time, by proving regularity under minimal measurability assumptions on time dependence.
• To demonstrate that solutions to SPDEs are infinitely differentiable in space locally, even when coefficients depend on time and randomness in a non-smooth way.
• To provide a robust method that avoids reliance on Malliavin calculus or global assumptions, using local Wentzell-type transformations to reduce SPDEs to deterministic equations.

Proposed method

• Apply Wentzell's random change of coordinates to eliminate stochastic terms in the SPDE, transforming it into a deterministic parabolic equation with random, time-dependent coefficients.
• Use the local Hörmander condition on the vector fields $\sigma^{d_1+k}_t$, $k=1,\dots,d_2$, to ensure the Lie algebra generates $\mathbb{R}^d$ in a ball $B$.
• Reduce the SPDE to a form where classical regularity results for parabolic equations can be applied, leveraging the fact that the transformed coefficients are uniformly nondegenerate.
• Construct a cutoff function $\zeta$ supported in $B_{R_0}$ and use it to localize the solution, ensuring the transformed equation satisfies the required integrability and smoothness conditions.
• Extend the solution beyond the terminal time $T$ using the heat equation with $H^m_2$-valued initial data, preserving regularity and enabling application of Theorem 2.3 in a larger time interval.
• Use a decomposition $u_t = v_t + w_t$, where $v_t$ handles initial data and $w_t$ satisfies a modified SPDE with zero initial condition, allowing reduction to the case of vanishing initial data.

Experimental results

Research questions

• RQ1 Can hypoellipticity be established for SPDEs with coefficients that are only measurable in time, without requiring Hölder or smooth time dependence?
• RQ2 Does the local Hörmander condition—on the Lie algebra generated by the noise vector fields—imply infinite differentiability of solutions even when coefficients are random and time-dependent?
• RQ3 Can Wentzell's method of random change of coordinates be adapted to SPDEs with non-Markovian, measurable coefficients to eliminate stochastic terms locally?
• RQ4 Is it possible to achieve local regularity for generalized solutions (in $\mathcal{D}^{-\infty}$) under minimal assumptions on time and randomness dependence?

Key findings

• Solutions to the SPDE are infinitely differentiable in space on any compact subset of $(s_1,s_2) \times B$ whenever the local Hörmander condition holds and the forcing terms $f_t, g^k_t$ are smooth.
• The method ensures that for any $\omega \in \Omega_0$, the solution $u_t$ is $C^\infty$ in $x$ on $B$ for $t \in (s_1,s_2)$, even if coefficients are only predictable in time.
• The key technical step is the local Wentzell transformation, which removes stochastic terms and reduces the problem to a deterministic parabolic equation with random, uniformly nondegenerate coefficients.
• The paper proves that the solution remains smooth even when the coefficients $\sigma^k_t$ are only measurable in $t$, a setting not covered by prior Malliavin calculus-based approaches.
• The result holds for generalized solutions in $\mathcal{D}^{-\infty}$, and the regularity is established locally, not globally, avoiding issues with growth at infinity.
• By extending the solution beyond $T$ via the heat equation and redefining coefficients, the method allows the application of Theorem 2.3 in a larger time interval, proving the full regularity result on $[S,T] \times B_r$.