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 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 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.     

Global existence of solutions for stochastic impulsive differential equations

In this paper we obtain some results on the global existence of solution to Itô stochastic impulsive differential equations in M([0,∞),? ⁿ ) which denotes the family of ? ⁿ -valued stochastic processes x satisfying sup_t∈[0,∞) \(\mathbb{E}\)|x(t)|² < ∞ under non-Lipschitz coefficients. The Schaefer fixed point theorem is employed to achieve the desired result. An example is provided to illustrate the obtained results.     

On existence,uniqueness and convergence of multi-valued stochastic diferential equations driven by continuous semimartingales

In this paper we study the existence and uniqueness of solutions of multi-valued stochastic diferential equations driven by continuous semimartingales when the coefcients are stochastically Lipschitz continuous.We also show the convergence results when the random coefcients or the diferentials converge.     

On existence,uniqueness and convergence of multi-valued stochastic differential equations driven by continuous semimartingales

In this paper we study the existence and uniqueness of solutions of multi-valued stochastic differential equations driven by continuous semimartingales when the coefficients are stochastically Lipschitz continuous. We also show the convergence results when the random coefficients or the differentials converge.     

On the existence and absence of global solutions of the first Darboux problem for nonlinear wave equations

For the one-dimensional wave equation with a power-law nonlinearity, we consider the first Darboux problem, for which we study issues related to the existence and absence of local and global solutions.