# Antoniouk A. Val.

### Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds

Antoniouk A. Val., Antoniouk A. Vict.

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1299–1316

We study the dependence on initial data for solutions of diffusion equations with globally non-Lipschitz coefficients on noncompact manifolds. Though the metric distance may not be everywhere twice differentiable, we show that, under certain monotonicity conditions on the coefficients and curvature of the manifold, there are estimates exponential in time for the continuity of a diffusion process with respect to initial data. These estimates are combined with methods of the theory of absolutely continuous functions to achieve the first-order regularity of solutions with respect to initial data. The suggested approach neither appeals to the local stopping time arguments, nor applies the exponential mappings on the tangent space, nor uses imbeddings of a manifold to linear spaces of higher dimensions.

### Nonexplosion and solvability of nonlinear diffusion equations on noncompact manifolds

Antoniouk A. Val., Antoniouk A. Vict.

Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1454–1472

We find sufficient conditions on coefficients of diffusion equation on noncompact manifold, that guarantee non-explosion of solutions in a finite time.
This property leads to the existence and uniqueness of solutions for corresponding stochastic differential equation with globally non-Lipschitz coefficients.

Proposed approach is based on the estimates on diffusion generator, that weakly acts on the metric function of manifold.
Such estimates enable us to single out a manifold analogue of monotonicity condition on the joint behaviour of the curvature of manifold and coefficients of equation.

### Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural?

Antoniouk A. Val., Antoniouk A. Vict.

Ukr. Mat. Zh. - 2006. - 58, № 8. - pp. 1011–1034

It is shown that the geometrically correct investigation of regularity of nonlinear differential flows on manifolds and related parabolic equations requires the introduction of a new type of variations with respect to the initial data. These variations are defined via a certain generalization of a covariant Riemannian derivative to the case of diffeomorphisms. The appearance of curvature in the structure of high-order variational equations is discussed and a family of a priori nonlinear estimates of regularity of any order is obtained. By using the relationship between the differential equations on manifolds and semigroups, we study $C^{∞}$ regular properties of solutions of the parabolic Cauchy problems with coefficients increasing at infinity. The obtained conditions of regularity generalize the classical coercivity and dissipation conditions to the case of a manifold and correlate (in a unified way) the behavior of diffusion and drift coefficients with the geometric properties of the manifold without traditional separation of curvature.

### Nonlinear-estimate approach to the regularity of infinite-dimensional parabolic problems

Antoniouk A. Val., Antoniouk A. Vict.

Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 579–596

We show how the use of nonlinear symmetries of higher-order derivatives allows one to study the regularity of solutions of nonlinear differential equations in the case where the classical Cauchy-Liouville-Picard scheme is not applicable. In particular, we obtain nonlinear estimates for the boundedness and continuity of variations with respect to initial data and discuss their applications to the dynamics of unbounded lattice Gibbs models.

### Higher-Order Relations for Derivatives of Nonlinear Diffusion Semigroups

Antoniouk A. Val., Antoniouk A. Vict.

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 117-122

We show that a special choice of the Cameron–Martin direction in the characterization of the Wiener measure via the formula of integration by parts leads to a set of natural representations for derivatives of nonlinear diffusion semigroups. In particular, we obtain a final solution of the non-Lipschitz singularities in the Malliavin calculus.