Nonlinear stochastic wave and heat equations

We study nonlinear wave and heat equations on ℝ ^d driven by a spatially homogeneous Wiener process. For the wave equation we consider the cases of d = 1, 2, 3. The heat equation is considered on an arbitrary ℝ ^d -space. We give necessary and sufficient conditions for the existence of a function-valued solution in terms of the covariance kernel of the noise. Received: 1 April 1998 / Revised version: 23 June 1999 / Published online: 7 February 2000     

On the large-scale structure of the tall peaks for stochastic heat equations with fractional Laplacian

Consider stochastic heat equations with fractional Laplacian on ${\mathbb{R}}^d$. The driving noise is generalized Gaussian which is white in time but spatially homogeneous. We study the large-scale structure of the tall peaks for (i) the linear stochastic heat equation and (ii) the parabolic Anderson model. We obtain the largest order of the peaks and compute the macroscopic Hausdorff dimensions of the peaks for (i) and (ii). These result imply that both (i) and (ii) exhibit multi-fractal behavior even though only (ii) is intermittent. This is an extension of a result of Khoshnevisan et al. (2017) to a wider class of stochastic heat equations.     

Derivatives of generalized farthest functions and existence of generalized farthest points

The relationship between directional derivatives of generalized farthest functions and the existence of generalized farthest points in Banach spaces is investigated. It is proved that the generalized farthest function generated by a bounded closed set having a one-sided directional derivative equal to 1 or −1 implies the existence of generalized farthest points. New characterization theorems of (compact) locally uniformly convex sets are given.     

On the existence of solutions of stochastic differential equations with singular drifts

Summary In this paper we prove the existence of solutions for a stochastic differential equation inR ^d, when the drift and the diffusion term are allowed to depend on a specific way on the local time of thedth coordinate of the process to be constructed. The methods of our construction are of purely probabilistic nature.     

On existence and uniqueness of solution of stochastic differential equations with heredity

We consider the Cauchy problem for the stochastic differential equation with the heredity where x _t(s) = x(s)for s?(- ∞,t).Existence and uniqueness theorems for the problem (1),(2)are proved inthe case,when instead of the Lipschitz condition for the functions a(t,u) and b(t,u)on u someless restrictive conditions (Ousgood or Hölder type)are satisfied, and the operator(Fx)(t) = x(t)-f(t,x _t) is invertible.Similar questions were considered in[1-4]     

On existence and uniqueness of solutions to uncertain backward stochastic differential equations简

This paper is concerned with a class of uncertain backward stochastic differential equations （UBSDEs） driven by both an m-dimensional Brownian motion and a d-dimensional canonical process with uniform Lipschitzian coefficients. Such equations can be useful in mod- elling hybrid systems, where the phenomena are simultaneously subjected to two kinds of un- certainties： randomness and uncertainty. The solutions of UBSDEs are the uncertain stochastic processes. Thus, the existence and uniqueness of solutions to UBSDEs with Lipschitzian coeffi- cients are proved.     

On the existence of an equivalent supermartingale density for a fork-convex family of stochastic processes

We prove that a fork-convex family $ \mathbb{W} $ of nonnegative stochastic processes has an equivalent supermartingale density if and only if the setH of nonnegative random variables majorized by the values of elements of $ \mathbb{W} $ at fixed instants of time is bounded in probability. A securities market model with arbitrarily many main risky assets, specified by the set $ \mathbb{W}\left( \mathbb{S} \right) $ of nonnegative stochastic integrals with respect to finite collections of semimartingales from an arbitrary indexed family S, satisfies the assumptions of this theorem.     

On the existence of weak solutions to stochastic differential equations with degenerate diffusion

For a certain class of stochastic differential equations with nonlinear drift and degenerate diffusion term existence of a weak solution is shown.     

On existence of solutions of multivalued stochastic differential equations with discontinuous coefficients

The subject of the paper is to find existence conditions of weak solutions to multivalued stochastic differential equations with discontinuous coefficients. First we prove that a non-exploding solution exists when the drift coefficient b satisfies linear growth and the diffusion coefficient σ is uniformly elliptic. On this basis, we continue to obtain a solution (up to the explosion time) in the weak sense under certain local integrability, improving the result of Rozkosz and S?omiński.     

On the existence of solutions with smooth density of stochastic differential equations in plane

In this paper we apply the Malliavin calculus for two-parameter Wiener functionals to show that the solutions of stochastic differential equations in plane have a smooth density under the restricted Hörmander's condition. This answers a question mentioned by Nualart and Sanz in [3].     

On almost global existence for nonrelativistic wave equations in 3D

Almost global solutions are constructed to three-dimensional, quadratically nonlinear wave equations. The proof relies on generalized energy estimates and a new decay estimate. The method applies to equations that are only classically invariant, such as the nonlinear system of hyperelasticity. © 1996 John Wiley & Sons, Inc.     

On existence of optimal solutions for stochastic differential equations and inclusions with current velocities

For a stochastic differential inclusion given in terms of current velocities (symmetric mean derivatives) on flat n-dimensional torus, we prove the existence of optimal solution minimizing a certain cost criterion. Then this result is applied to the problem of optimal control for equations with current velocities.     

On the existence of product stochastic measures

In this paper, we prove that the existence of product stochastic measures depends on the axiom-system of set theory: If one accepts the axiom of choice, the answer is negative, and we give a counter-example where the product stochastic measure doesn't exist; but in the Solovay model (one kind of set theory which refuses the axiom of choice), the answer is positive, and we give a proof.