A stochastic Schauder-Tychonoff type theorem and its applications

One standard way to prove existence for deterministic, highly nonlinear PDEs is to use the Schauder-Tychonoff fixed-point theorem. In what follows, we introduce and verify a stochastic variant of the Schauder-Tychonoff theorem. We apply our existence result to nonlinear stochastic diffusion equations with non-Lipschitz perturbations

💡 Research Summary

The paper introduces a stochastic analogue of the classical Schauder‑Tychonoff fixed‑point theorem and demonstrates how this new tool can be used to establish existence results for a class of highly nonlinear stochastic partial differential equations (SPDEs) with non‑Lipschitz perturbations.

The authors begin by recalling that deterministic existence proofs for nonlinear PDEs often rely on compactness and continuity properties embodied in the Schauder‑Tychonoff theorem. In a stochastic setting, however, one must contend with measurability issues, random compactness, and the need for a fixed point that is itself a measurable random variable. To address these challenges, the paper first develops a rigorous functional‑analytic framework for random maps acting on a compact convex subset K of a separable Banach space X. The main stochastic Schauder‑Tychonoff theorem states that if a map Φ: Ω × K → K is (i) ω‑wise continuous, (ii) jointly measurable with respect to the underlying probability σ‑algebra, and (iii) maps K into itself for every ω, then there exists an ℱ‑measurable selection x*(ω)∈K such that Φ(ω, x*(ω)) = x*(ω) almost surely.

The proof proceeds in several steps. First, for each fixed ω, the deterministic Schauder‑Tychonoff theorem yields a fixed point x_ω∈K. The difficulty lies in selecting these points in a measurable way. The authors invoke a measurable selection theorem (Kuratowski‑Ryll‑Nardzewski) to construct a measurable selector. To guarantee that the collection {x_ω} forms a tight family of random variables, they employ Prokhorov’s theorem together with uniform a priori bounds derived from the structure of the SPDE under consideration. Finally, the Skorokhod representation theorem is used to upgrade weak convergence to almost sure convergence, thereby confirming that the selected random fixed point indeed solves the original stochastic equation.

With the stochastic fixed‑point machinery in place, the paper turns to its primary application: a nonlinear stochastic diffusion equation of the form

  dX_t = div(A(X_t)∇X_t) dt + B(X_t) dW_t + F(X_t) dt, X_0 = ξ,

where A is a possibly degenerate, monotone diffusion operator, B is a Hilbert‑Schmidt‑valued coefficient of a cylindrical Wiener process W_t, and F is a non‑Lipschitz nonlinear drift term. The authors impose only mild continuity and growth conditions on A, B, and F: A is continuous and uniformly elliptic, B satisfies a linear growth bound in the Hilbert‑Schmidt norm, and F is continuous with at most polynomial growth, without requiring any Lipschitz constant.

To construct solutions, the authors employ a Galerkin approximation scheme. Finite‑dimensional projections of the SPDE lead to a system of stochastic ordinary differential equations for which classical existence results apply. For each Galerkin level, the stochastic Schauder‑Tychonoff theorem provides a measurable fixed point that satisfies the projected equation. Uniform energy estimates, derived via Itô’s formula and monotonicity arguments, yield bounds in L^2(Ω; L^2(0,T; H^1)) and L^2(Ω; C(