A probability measure μ on Rd is said to satisfy the log-Sobolev inequality if for every smooth compactly supported function f:Rd→R , the entropy of  f2 , which by definition equals Entμf2=∫Rdf2logf2dμ−(∫Rdf2dμ)log(∫Rdf2dμ), possesses a bound (1) Entμf2≤2c∫Rd‖∇f‖2dμ with some constant c. The least possible constant c such that (1) holds for every compactly supported smooth f is called the log-Sobolev constant for the measure μ; the multiplier 2 in (1) is chosen in such a way that for the standard Gaussian measure on Rd , its log-Sobolev constant equals 1.

The weighted log-Sobolev inequality has the form (2) Entμf2≤2∫Rd‖W∇f‖2dμ, where the function W, taking values in Rd×d , has the meaning of a weight. Clearly, one can consider (1) as a particular case of (2) with constant weight W equal to c multiplied by the identity matrix. The problem of giving explicit conditions on μ that ensure the log-Sobolev inequality or its modifications is intensively studied in the literature, in particular, because of numerous connections between these inequalities with measure concentration, semigroup properties, and so on (see, e.g., [8]). Motivated by this general problem, in this paper, we propose an approach that is based mainly on martingale methods and provides explicit bounds for the entropy with the right-hand side given in a certain integral form.

Our approach is motivated by the well-known fact that, on a path space of a Brownian motion, the log-Sobolev inequality possesses a simple proof based on fine martingale properties of the space (cf. [1, 6]). We observe that a part of this proof is, to a high extent, insensitive w.r.t. the structure of the probability space; we formulate a respective martingale bound for the entropy in Section 1.1. To apply this general bound on a probability space of the form (Rd,μ) , one needs a proper martingale structure therein. In Section 2, we introduce such a structure in terms of a trimming filtration, defined in terms of a set of trimmed regions in Rd . This leads to an integral bound for the entropy on (Rd,μ) . In Section 3, we show the way how this bound can be used to obtain a weighted log-Sobolev inequality; this is made in the one-dimensional case d=1 , although we expect that similar arguments should be effective for the multidimensional case as well; this is a subject of our further research.

Let μ be a probability measure on Rd with Borel σ-algebra B(Rd) . Our further aim is to apply the general martingale bound from Theorem 1 in the particular setting (Ω,F,P)=(Rd,B(Rd),μ) . To this end, we first construct a filtration {Ft,t∈[0,1]} .

In what follows, we denote Nμ={A∈F:μ(A)=0} (the class of μ-null Borel sets).

Fix the family {Dt,t∈[0,1]} of closed subsets of Rd such that: (i)

Ds⊂Dt,s≤t ;

By the construction, F={Ft} is complete. It is also clear that, by property (ii) of the family {Dt} , the σ-algebra F0 is degenerate and, by property (iii), the filtration F is continuous. Hence, we can apply Theorem 1.

Fix a Borel-measurable function g:Rd→R+ that is square-integrable w.r.t. μ. Consider it as a random variable on (Ω,F,P)=(Rd,B(Rd),μ) and define gt=E[g|Ft],t∈[0,1]. Since the σ-algebra possesses an explicit description, we can calculate every gt directly; namely, for t>0 0$]]> and μ-a.a. x, we have (8) gt(x)={g(x),x∈Dt,Gt,x∈Qt, where we denote (9) Gt=1μ(Qt)∫Qtg(y)μ(dy). Note that μ(Qt)>0 0$]]> for t<1 and the function G:[0,1)→R+ is continuous. In what follows, we consider the modification of the process {gt} defined by (8) for every x∈Rd . Its trajectories can be described as follows. Denote (10) τ(x)=inf{t:x∈Dt}; then by property (iii) of the family {Dt} we have τ(x)=min{t:x∈Dt} , and by property (ii) we have τ(x)<1,x∈Rd,τ(x)=0⇔x∈D0 . Then, for a fixed x∈Rd , we have gt(x)=g(x)1t≥τ(x)+Gt1t<τ(x),t∈[0,1], which is a càdlàg function because {Gt} is continuous on [0,1) . Theorem 2.

Let g be a Borel-measurable function g:Rd→R+ , square-integrable w.r.t. μ. Let {Dt} be a family of trimmed regions that satisfy (i)–(iii).

Then Entμg≤∫Rd(g(x)−Gτ(x))2Gτ(x)μ(dx), where the functions G and τ are defined by (9) and (10), respectively.

Let us continue with the proof of (11). Assume first that g≥c with some c>0 0$]]>. Then gt≥c , and consequently the process 1/gt− is left continuous and bounded. In addition, the function Gt=It/μt is bounded on every segment [0,T]⊂[0,1) .

Fix T<1 and take a sequence {λn} of dyadic partitions of [0,T] , λn={tkn,k=0,…,2n},tkn=Tk2n, and define gtn=g01t=0+ ∑k=12ngtk−1n1t∈(tk−1n,tkn]. For every fixed t>0 0$]]>, the value gtn equals the value of g at some (dyadic) point tn<t , and tn→t− . Hence, 1gtn→1gt−,n→∞, pointwise. In addition, because of the additional assumption g≥c , this sequence is bounded by 1/c . Hence, by the dominated convergence theorem, E∫0T1gt−d⟨g⟩t=limn→∞E ∑k=12n1gtk−1n(⟨g⟩tkn−⟨g⟩tk−1n); here we take into account that the point t=0 in the left-hand side integral is negligible because gt→Eg,t→0+ , in L2 , and consequently ⟨g⟩t→0 , t→0+ , in L1 . By Lemma 1, E ∑k=12n1gtk−1n(⟨g⟩tkn−⟨g⟩tk−1n)=E ∑k=12n∫Dtkn∖Dtk−1n(g(x)−Gτ(x))2Gtk−1nμ(dx); recall that gtk−1n(x)=Gtk−1n for x∉Dtk−1n . Next, for x∈Dtkn∖Dtk−1n , we have |τ(x)−tk−1n|≤2−n . Because Gt,t∈[0,T] , is uniformly continuous and separated from zero, and Gτ(x),x∈DT is bounded, we obtain that E∫0T1gt−d⟨g⟩t=limn→∞E ∑k=12n∫Dtkn∖Dtk−1n(g(x)−Gτ(x))2Gtk−1nμ(dx)=∫DT(g(x)−Gτ(x))2Gτ(x)μ(dx). Taking T→1− and applying the monotone convergence theorem to both sides of the above identity, we get (11).

To remove the additional assumption g≥c , consider the family gtn=gt+1/n . Then ⟨gn⟩=⟨g⟩ , gn(x)−Gτ(x)n=g(x)−Gτ(x) , gt−n=gt−+(1/n),Gτ(x)n=Gτ(x)+(1/n) . Hence, we can write (11) for gn , apply the monotone convergence theorem to both sides of this identity, and get (11) for g.  □

In this section, we show the way how the integral bound for the entropy, established in Theorem 2, can be used to obtain weighted log-Sobolev inequalities. Consider a continuous probability measure μ on (R,B(R)) and denote by pμ the density of its absolutely continuous part. Fix a family of segments Dt=[at,bt],t∈[0,1) , where a0=b0 , the function at is continuous and decreasing to −∞ as t→1− , and the function b· is continuous and increasing to +∞ as t→1− . Then the family Dt=[at,bt],t∈[0,1),D1=R, satisfies the assumptions imposed before. Hence, Theorem 2 is applicable.

We call a function f:R→R symmetric w.r.t. the family {Dt} if f(at)=f(bt),t∈[0,1). In the following proposition, we apply Theorem 2 to g=f2 , where f is smooth and symmetric.