Abstract
We provide a complete elaboration of the L^2-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics with multiplicative noise, studying the longtime behavior of the strongly continuous contraction semigroup solving the abstract Cauchy problem for the associated backward Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in the corresponding Fokker–Planck framework and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. We extend these results to weakly differentiable diffusion coefficient matrices, introducing multiplicative noise for the corresponding stochastic differential equation. The rate of convergence is explicitly computed depending on the choice of these coefficients and the potential giving the outer force. In order to obtain a solution to the abstract Cauchy problem, we first prove essential self-adjointness of non-degenerate elliptic Dirichlet operators on Hilbert spaces, using prior elliptic regularity results and techniques from Bogachev, Krylov and Röckner. We apply operator perturbation theory to obtain essential m-dissipativity of the Kolmogorov operator, extending the m-dissipativity results from Conrad and Grothaus. We emphasize that the chosen Kolmogorov approach is natural, as the theory of generalized Dirichlet forms implies a stochastic representation of the Langevin semigroup as the transition kernel of a diffusion process which provides a martingale solution to the Langevin equation with multiplicative noise. Moreover, we show that even a weak solution is obtained this way.
Highlights
We study the exponential decay to equilibrium of Langevin dynamics with multiplicative noise
The approach we use here was introduced algebraically by Dolbeault, Mouhot and Schmeiser and made rigorous including domain issues in [4] by Grothaus and Stilgenbauer, where it was applied to show exponential convergence to equilibrium of a Fiber laydown process on the unit sphere. This setting was further generalized by Wang and Grothaus in [5], where the coercivity assumptions involving in part the classical Poincaré inequality for Gaussian measures were replaced by weak Poincaré inequalities, allowing for more general measures for both the spatial and the velocity
Since (BT, D(T )) is densely defined, by Lemma 2.1 (i) and (ii), it is closable with bounded and its closure (BT) = (BT )∗∗, which is a bounded operator on H with the stated norm
Summary
We study the exponential decay to equilibrium of Langevin dynamics with multiplicative noise. The corresponding evolution equation is given by the following stochastic differential equation on R2d , d ∈ N, as d Xt = Vt dt,. D Vt = b(Vt )dt − ∇Φ(Xt )dt + 2σ (Vt )d Bt , where Φ : Rd → R is a suitable potential whose properties are specified later, B = (Bt )t≥0 is a standard d-dimensional Brownian motion, σ : Rd → Rd×d a variable. Mathematics Subject Classification: 37A25, 47D07, 35Q84, 47B44, 47B25 Keywords: Langevin equation, Multiplicative noise, Hypocoercivity, Essential m-dissipativity, Essential self-adjointness, Fokker–Planck equation
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