Pierre-A. Vuillermot

On the behavior of solutions to certain parabolic SPDE's driven by wiener processes

In this article we prove new results concerning the structure and the stability properties of the global attractor associated with a class of nonlinear stochastic partial differential equations driven by finite-dimensional Wiener processes. This class encompasses important equations that occur in the mathematical analysis of certain migration phenomena in population dynamics and population genetics. The solutions to such equations are generalized random fields whose long-time behavior we investigate in detail. In particular, we unveil the mechanism whereby these random fields approach the global attractor by proving that their asymptotic behavior is entirely controlled by that of their spatial average. We also show how to determine explicitly the corresponding Lyapunov exponents when the nonlinearities of the noise-term of the equations are subordinated to the nonlinearity of the drift-term in some sense. The ultimate picture that emerges from our analysis is one that displays a phenomenon of exchange of stability between the components of the global attractor. We provide a very simple interpretation of this phenomenon in the case of Fishers equation of population genetics. Our method of investigation rests upon the use of martingale arguments, of a comparison principle and of some simple ergodic properties for certain Lebesgue- and Ito integrals.

Long-time behavior of solutions to a class of quasilinear parabolic equations with random coefficients

Abstract In this article we prove new results concerning the long-time behavior of random fields that are almost surely solutions to a class of stochastic parabolic Neumann problems defined on open bounded connected subsets of R N. Under appropriate ellipticity and regularity hypotheses, we first prove that every such random field stabilizes almost surely in a suitable topology around a spatially homogeneous random process whose statistical properties are entirely determined by those of the given coefficients in the equations. In addition, when the coefficients of the lower-order terms in the equations are stationary random processes, the nature of the equations that we investigate leads us to consider two complementary situations according to whether the average of those processes is zero or not. If their average is different from zero and if the processes are ergodic, we prove that every random field stabilizes almost surely and exponentially rapidly in the uniform topology around a spatially and temporally homogeneous asymptotic state, which depends only on the sign of the average. In this case we can also determine the corresponding Liapunov exponents exactly. In contrast, if the average of the processes is equal to zero we need more structure to identify the asymptotic states properly. The cases where the coefficients of the lower-order terms in the equations are either stationary random processes whose statistics are governed by the central limit theorem, or Gaussian processes that share some of the features of the Ornstein-Uhlenbeck process, are of special interest and we investigate them in detail. In all cases we can also provide estimates for the average time that the random fields spend in small neighborhoods of the asymptotic states. Our methods of proof rest chiefly upon the use of parabolic comparison principles.

Lyapunov exponents and stability for nonlinear SPDE's driven by finite-dimensional Wiener processes

Abstract We present new results concerning the stability properties of the global attractor associated with a class of nonlinear SPDEs driven by finite-dimensional Wiener processes of arbitrary covariance. In particular, we show how to determine explicitly certain Lyapunov exponents when the nonlinearities of the noise-terms of the equations are subordinated to the nonlinearities of the drift-terms in some sense. Our method of investigation rests upon the use of a comparison principle and of simple ergodic properties for certain Ito martingales

Large-time asymptotic equivalence for a class of non-autonomous semilinear parabolic equations

Abstract In this article we prove new results concerning the long-time behavior of solutions to a class of non-autonomous semilinear parabolic Neumann boundary-value problems defined on open bounded connected subsets Ω of RN. The nature of the equations that we investigate leads us to consider two complementary situations, according to whether the time-dependent lower order terms in the equations possess recurrence properties. If the lower order terms are recurrent, we prove that every solution stabilizes around a spatially homogeneous and recurrent solution of the same Neumann problem in the C1 ( Ω )-topology. In contrast, if the lower order terms are not recurrent, the asymptotic states need not be solutions to the original problem and we prove that every solution stabilizes around such an asymptotic state again in the C1 ( Ω )-topology. In all cases the dynamics of the asymptotic solutions are governed by a compact and connected set of scalar ordinary differential equations, which are thereby asymptotically equivalent to the original Neumann problem for large times. A major difficulty to be bypassed in the proofs of our theorems stems from the fact that we allow the nonlinearitics to depend explicitly on the gradient of the unknown function. Our method of proof rests upon the use of comparison principles and upon the existence of exponential dichotomies for the family of evolution operators associated with the principal part of the equations. It is also based on ideas that stem for the classic reduction methods for non-autonomous finite-dimentional dynamical systems originally devised by Miller, Strauss-Yorke and Sell .

Smooth manifolds for semilinear wave equations on R2: On the existence of almost-periodic breathers

Abstract In this paper, we prove several nonexistence and existence results for certain real solutions to semilinear wave equations in one space dimension. These solutions represent time almost-periodic free vibrations which decay toward a constant equilibrium solution as the spatial variable goes to infinity. Our nonexistence results are of two kinds. We first prove that the requirements of almost periodicity in time and of spatial decay at infinity force such solutions to be trivial, whenever their spectrum does not interact with the nonlinearity in a certain sense. We also prove non-existence results without requiring any spectral conditions, but this is at the expense of having to impose more stringent conditions on the nonlinearities, such as certain convexity properties in the vicinity of the equilibrium solution. Finally, we prove an existence result for time almost-periodic free vibrations whose profiles decay exponentially rapidly toward a constant equilibrium solution. To accomplish this, we convert the wave equation into a dynamical system in which the original spatial variable plays the role of time; we then embed that dynamical system into an appropriate Banach space of time almost-periodic functions, so that the one-parameter family of stable and unstable manifolds which we construct carry the solutions that we seek. The major difficulty to overcome in our construction is a small divisor problem; it is related to the fact that the spectrum of the infinitesimal generator for the linearized flow is a pure point spectrum without gaps. We discuss several examples and we also stress some important qualitative differences which distinguish the almost-periodic case from the purely periodic one.

On the large-time dynamics of a class of parabolic equations subjected to homogeneous white noise: Itô's case

Abstract We present new results concerning the long-time behavior of random fields that are solutions in some variational sense to a class of semilinear parabolic equations subjected to homogeneous white noise. Our main results state that these random fields eventually converge to a random global attractor. This attractor is represented by a single random variable which, with probability one, takes on its values in a two element-set consisting of spatially and temporally homogeneous asymptotic states. We can then completely elucidate the behavior of the random fields around it.

Forward-Backward Stochastic Differential Equations Generated by Bernstein Diffusions

In this article, we prove new results regarding hitherto unknown relations that exist between certain Bernstein diffusions on the one hand and processes that typically occur in forward-backward systems of stochastic differential equations on the other hand. More specifically, we consider Bernstein diffusions that can wander in bounded convex domains where d is arbitrary, and which are generated there by a forward-backward system of decoupled linear deterministic parabolic partial differential equations. This makes them reversible Itô diffusions under some conditions that pertain to their marginal distributions, which then allows us to construct D × d ×d2 -valued processes that are weak solutions to suitably defined forward -backward systems of coupled stochastic differential equations. Moreover, we also consider the converse problem, namely, that of knowing whether the first component of a weak solution to a given forward-backward system is a Bernstein diffusion in some sense, which we solve affirmatively in a specific case.

A class of elliptic partial differential equations with exponential nonlinearities (Corrigendum)

We wish to correct a faulty argument that slipped into the proof of Step 5 of Proposition 3.1 of [1] (the Palais-Smale Property). The dominated convergence argument following relation (77) on p. 512 may not be applied as indicated. Indeed, the integer _~ =N(~, x) depends on x in general so that it is not clear whether Zi ~ Er,,p,. A correct argument which also provides a simplification may be based on the inequality

On some Gaussian Bernstein processes in and the periodic Ornstein–Uhlenbeck process

ABSTRACT In this article, we prove new results regarding the existence of Bernstein processes associated with the Cauchy problem of certain forward–backward systems of decoupled linear deterministic parabolic equations defined in Euclidean space of arbitrary dimension , whose initial and final conditions are positive measures. We concentrate primarily on the case where the elliptic part of the parabolic operator is related to the Hamiltonian of an isotropic system of quantum harmonic oscillators. In this situation there are many Gaussian processes of interest whose existence follows from our analysis, including N-dimensional stationary and non-stationary Ornstein–Uhlenbeck processes, as well as Bernstein bridges which may be interpreted as Markovian loops in a particular case. We also introduce a new class of stationary non-Markovian processes, which we eventually relate to the N-dimensional periodic Ornstein–Uhlenbeck process, which is generated by a one-parameter family of non-Markovian probability measures. In this case, our construction requires the consideration of an infinite hierarchy of pairs of forward–backward heat equations associated with the pure point spectrum of the elliptic part, rather than just one pair in the Markovian case. We finally stress the potential relevance of these new processes to statistical mechanics, the random evolution of loops, and general pattern theory.

Remarks on Some Strongly Nonlinear Degenerate Sturm-Liouville Eigenvalue Problems

Publisher Summary This chapter discusses a new theorem concerning the existence and the C 1,1 -regularity of countably many eigensolutions for a class (SL) of strongly nonlinear degenerate Sturm–Liouville eigenvalue problems with Young function nonlinearities in their principal part. The basic idea inherent in the proof of the theorem is threefold: For each ɛ ∈ (0, l), the problem (SL) is replaced first by a class (SL)ɛ of regularized, nondegenerate Sturm–Liouville eigenvalue problems with strictly convex nonlinearities. The monotonicity and compactness arguments are then invoked to infer that problem (SL)ɛ possesses at least countably many C (2) -eigensolutions. The existence and the C 1,1 -regularity of countably many eigen-solutions to problem (SL) are finally achieved by taking the ɛ0-limit in problem (SL)ɛ. The necessary bounds uniform in ɛ required to do so are obtained through new convexity inequalities that characterize the shape of the given nonlinearities. Two examples and a counter-example are given that illustrate the role of the various hypotheses, and which show that C l,1 -regularity is a nearly optimal result.