Stochastic Differential Inclusions

• Stochastic Differential Inclusions are evolution equations with set-valued drifts and stochastic perturbations, enabling the analysis of systems with constraints and non-smooth dynamics.
• They employ variational methods, generalized Yosida approximations, and projection schemes to establish existence, uniqueness, and convergence of numerical solutions.
• Applications span from sweeping processes in constrained dynamics to stochastic optimization and learning, impacting fields such as SPDE analysis and crowd motion.

Stochastic differential inclusions (SDIs) are evolution equations in which the drift or dynamics are described by set-valued maps, and stochastic perturbations—typically via Brownian motion or more general stochastic processes—are present. These systems generalize both stochastic differential equations (SDEs) and deterministic differential inclusions, enabling the modeling of stochastic phenomena subject to possibly non-smooth constraints, multivalued nonlinearities, state-dependent discontinuities, or time-varying constraint sets. SDIs have become an essential mathematical tool for the rigorous analysis of systems in stochastic control, optimization, learning algorithms, singular and degenerate SPDEs, and applications with constraints or discontinuities.

1. Foundational Frameworks and Definitions

The prototypical form of a stochastic differential inclusion is

$dX_t \in F(t, X_t)\, dt + \Sigma(t, X_t)\, dB_t,$

where $$F : [0,T] \times \mathbb{R}^d \rightrightarrows \mathbb{R}^d$$ is a set-valued (multifunction) map—often maximal monotone, upper semicontinuous, or of Clarke subdifferential type—and $\Sigma$ is the diffusion coefficient, with $B_t$ a Brownian motion (or other stochastic process). The solutions are typically required to satisfy a “measurable selection” condition: almost surely, the drift at each time remains within the set prescribed by F.

Stochastic evolution inclusions (SEIs) extend this framework to infinite-dimensional settings (e.g., Gelfand triples $V \subset H \subset V^*$), relevant for SPDEs with set-valued drift or constraints (Gess et al., 2011, Fan et al., 17 Feb 2025). The notion of solution can be in the strong (pathwise), weak (in law), or variational sense, with limits defined via upper hemicontinuity, closure in measure, or relaxation when approximations are invoked.

A key instance is the stochastic sweeping process, expressed as

$$dX_t + N(C(t), X_t) \ni f(t, X_t)\,dt + \sigma(t, X_t)\,dB_t,$$

where $N(C(t), X_t)$ is the (proximal) normal cone to the (possibly non-convex, time-dependent) moving set $C(t)$ (Bernicot et al., 2010). This formulation natively incorporates instantaneous constraints and is central in contact and crowd dynamics.

2. Well-Posedness, Existence, and Uniqueness

The existence and uniqueness of solutions to SDIs fundamentally rely on the structure of the set-valued drift and the regularity of the stochastic perturbation. For SDIs driven by maximal monotone operators $A$, equipped with monotonicity, weak coercivity, and upper semicontinuity properties, variational techniques based on Yosida approximation, compactness, and monotonicity methods are standard (Gess et al., 2011, Fan et al., 17 Feb 2025).

For multi-valued operators in Banach/Gelfand triples, the generalized (possibly nonlinear) Yosida approximation

$$A_\lambda(x) := \frac{1}{\lambda} J(x - R_\lambda(x)),$$

with $J$ the duality mapping and $R_\lambda$ the nonlinear resolvent, provides strong approximations for constructing mild solutions and ensuring pathwise uniqueness (Fan et al., 17 Feb 2025). The classical resolvent operator is encompassed as a special case.

In the context of sweeping processes with stochastic perturbations, well-posedness (existence and pathwise uniqueness) is established under the following assumptions (Bernicot et al., 2010):

• The constraint set-valued map $t \mapsto C(t)$ is uniformly prox-regular, admissible (“inward pointing”), and Lipschitz in the Hausdorff metric.
• The drift and diffusion terms $f(t, x),\ \sigma(t,x)$ are bounded and Lipschitz in the spatial variable.
• For suitably regular $C(t)$ (controlled by a function $\Phi$), existence and uniqueness hold in $L^4(\Omega; L^\infty([0,T]))$.

If the regularity of $C(t)$ is weakened, uniqueness may still be concluded in weaker solution spaces.

For SEIs in Hilbert spaces perturbed by both additive and multiplicative Wiener noise, existence and uniqueness (variational solutions) are proven using monotonicity, weak coercivity, and compactness, together with approximation by finite-dimensional/subspace truncations (Gess et al., 2011).

3. Numerical Approximation: Euler–Catching-Up Schemes

Discretization of SDIs, especially sweeping processes, typically deploys an Euler or Moreau’s “catching-up” scheme adapted to the set-valued drift with stochastic increments (Bernicot et al., 2010). For a partition $0 = t_0 < t_1 < ... < t_n = T$, and a sample path of the driving Brownian motion, the discrete approximation is as follows:

• For $t \in [t_k, t_{k+1})$,

$$X_h(t) = P_{C(t_{k+1})}\big[ X_h(t_k) + h\,f(t_k, X_h(t_k)) + \sigma(t_k, X_h(t_k)) (B_t - B_{t_k}) \big],$$

where $P_{C}$ denotes the Euclidean projection onto $C$.

The main convergence theorem asserts that as $h \to 0$, these discretizations converge (in $L^\infty$ norm, almost surely in time for each realization) to the unique solution of the original SDI, provided the constraint sets retain uniform prox-regularity and the drift/diffusion map conditions are met.

Key analytic tools in the proof include:

• Itô’s formula for $C^2$ control functions, Gronwall-type inequalities, and stochastic estimates (Doob’s, BDG).
• Careful control of the time-dependence of the constraint, via estimates ensuring that the variation in $C(t)$ remains tractable.

4. Singular and Multi-Valued Operators: Evolution Inclusions

For evolution inclusions where the drift is highly singular or genuinely multi-valued—such as the stochastic total variation flow and fast diffusion-type SPDEs—general variational techniques have been developed (Gess et al., 2011, Fan et al., 17 Feb 2025). The typical equation is

$dX(t) + A(X(t)) \,dt \ni B(X(t))\,dW(t),$

with $A$ maximal monotone (but possibly degenerate or non-coercive), and $B$ possibly state-dependent (multiplicative noise).

In these settings:

• Existence and uniqueness are secured via nonlinear Yosida regularization, compactness, and weak convergence arguments.
• Examples include stochastic p-Laplace equations ($p \in [1,2)$), stochastic porous medium equations (with $A = \Delta \Phi$, $\Phi$ maximal monotone), and plasma diffusion equations ($A$ with logarithmic or non-classical nonlinearities).
• For additive noise (state-independent $B$), the Markov semigroup associated with the solution process possesses a unique invariant measure and exhibits weak-* ergodicity under suitable dissipativity, as shown for the stochastic total variation flow.

A notable property for certain multi-valued SEIs (e.g., with linear multiplicative noise) is finite-time extinction: almost all trajectories reach zero in finite time, a consequence of the interplay between operator degeneracy and noise structure (Fan et al., 17 Feb 2025).

5. Applications: From Constrained Dynamics to Distributed Optimization

Stochastic differential inclusions naturally arise in a variety of applied contexts:

• Granular flows and crowd motion: The feasible non-overlap configurations of rigid bodies (e.g., disks) define a time-dependent constraint set, with dynamics modeled as a stochastic sweeping process subject to random perturbations (modeling, e.g., panic or fluctuations in velocity) (Bernicot et al., 2010).
• Stochastic optimization and learning: Evolution inclusions underpin stochastic gradient-type algorithms with projection or non-smooth constraints, approximate drift, and state-dependent randomness. The generalization of the Borkar–Meyn theorem to inclusions permits convergence analysis even when errors or approximation yield set-valued mean fields (Ramaswamy et al., 2015).
• Stochastic recursive inclusions with Markov noise: Average-case (invariant measure) and controlled cases are rigorously handled by averaging the set-valued drift over the stationary distributions of the noise process, leading to well-posed asymptotic pseudotrajectories (Yaji et al., 2016, Yaji et al., 2016).
• Non-instantaneous impulses and delay systems: SDIs with impulses or delay, modeled via Clarke’s subdifferential, admit mild solutions when formulated in Hilbert spaces using evolution operators and fixed-point methods, and can account for realistic physical models with memory and impulsive events (Upadhyay et al., 2021).

6. Mathematical and Methodological Advances

The theoretical development of SDIs leverages and extends several key methodologies:

• Monotonicity and maximal monotonicity: Ensures the existence of solutions for possibly degenerate operators and is critical for the construction and convergence of generalized Yosida approximations (Gess et al., 2011, Fan et al., 17 Feb 2025).
• Regularization and approximation: Nonlinear Yosida approximation, designed via duality mappings in Banach spaces, provides flexibility in handling more general coercivity conditions and variational frameworks beyond classical (linear) structure.
• Pathwise and variational solutions: Variational techniques based on Bochner spaces are essential for infinite-dimensional or singular/degenerate problems, with “mild” or integrated solution concepts accommodating non-smooth, impulsive, or delay terms.
• Analysis of ergodicity and extinction: Employing Lyapunov functions, dissipativity estimates, and covariance inequalities to establish ergodicity, invariant measures, and extinction behavior, even for entirely multi-valued and singular systems.

7. Extended Examples and Model Classes

A representative selection of concrete examples and model classes for which the established theory applies:

Model Type	Operator Structure	Noise Type
Stochastic sweeping process with moving constraints	Proximal normal cone	Additive, multiplicative
Stochastic total variation flow (1-Laplacian)	Subgradient (p = 1)	Additive/state-dependent
Stochastic p-Laplace (p ∈ [1,2))	Maximal monotone	Additive/multiplicative
Plasma diffusion (logarithmic)	Singular nonlinear	Additive/multiplicative
Stochastic porous media equation	Monotone nonlinearity	Additive/multiplicative
Rigid body/crowd motion	Proximal normal cone	Additive/Lipschitz
Delay with non-instantaneous impulses	Clarke's subdifferential	Multiplicative (Rosenblatt process)

Each class requires tailored conditions on the operators, growth, regularity, and the structure of noise, but all fit within the general framework provided by the variational/stochastic analysis of multi-valued inclusions.

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In summary, stochastic differential inclusions encompass a broad, rigorously defined class of evolution equations where stochasticity, set-valued/non-smooth dynamics, and potentially moving or time-dependent constraints interact. Foundational results now guarantee well-posedness, convergence of discretization schemes, and characterization of invariant measures under broad conditions, with wide-reaching applications to singular SPDEs, constrained stochastic dynamics, and modern stochastic optimization. Recent advances—including nonlinear Yosida approximation, treatment of pathwise uniqueness, and extinction phenomena—demonstrate the flexibility and depth of the theory in addressing challenging problems at the intersection of stochastic analysis, variational methods, and dynamical systems.